PHYSICS 11MIDTERM LECTURE NOTES

COVERAGE1. Mechanical Equilibrium

a. Forceb. Equilibriumc. Support Forced. Equilibrium of Moving Objectse. Vectors

2. Newton’s First Law of Motiona. Aristotle on Motionb. Copernicus and the Moving Earthc. Galileo on Motiond. Newton’s Law of Inertiae. Mass a Measure of Inertiaf. The Moving Earth

3. Linear Motiona. Motion is Relativeb. Speedc. Velocity

d. Acceleratione. Free Fallf. Graphs of Motion

4. Projectile Motiona. Velocity Vectorsb. Components of Vectorsc. Projectile Motionc. Projectiles Launched Horizontallyd. Projectiles Launched at an Angle

5. Newton’s Second Law of Motiona. Force Causes Accelerationb. Mass Resist Accelerationc. Newton’s Second Lawd. Frictione. Applying Force - Pressuref. Falling and Air Resistance

Mechanical EquilibriumAn object in mechanical equilibrium is stable, without changes in motion. Things that are in balance with one another illustrate equilibrium.Things in mechanical equilibrium are stable, without changes of motion. An unbalanced external force would be needed to change their resting state. “A force is needed to change an object’s state of motion.” ForceNet ForceA force is a push or a pull. A force of some kind is always required to change the state of motion of an object. The combination of all forces acting on an object is called the net force. The net force on an object changes its motion.The scientific unit of force is the newton, abbreviated N.The net force depends on the magnitudes and directions of the applied forces.Tension and WeightA stretched spring is under a “stretching force” called tension. Pounds and newtons are units of weight, which are units of force. The upward tension in the string has the same magnitude as the weight of the bag, so the net force on the bag is zero. The bag of sugar is attracted to Earth with a gravitational force of 2 pounds or 9 newtons. There are two forces acting on the bag of sugar:

• tension force acting upward • weight acting downward

The two forces on the bag are equal and opposite. The net force on the bag is zero, so it remains at rest.Mechanical EquilibriumYou can express the equilibrium rule mathematically as F = 0. Mechanical equilibrium is a state wherein no physical changes occur. Whenever the net force on an object is zero, the object is in mechanical equilibrium—this is known as the equilibrium rule.For a suspended object at rest, the forces acting upward on the object must be balanced by other forces acting downward.The vector sum equals zero. think!If the gymnast hangs with her weight evenly divided between the two rings, how would scale readings in both supporting ropes compare with her weight? Suppose she hangs with slightly more of her weight supported by the left ring. How would a scale on the right read?Answer: In the first case, the reading on each scale will be half her weight. In the second case, when more of her weight is supported by the left ring, the reading on the right reduces to less than half her weight. The sum of the scale readings always equals her weight.

What forces act on a book lying at rest on a table? • One is the force due to gravity—the weight of the book.• There must be another force acting on it to produce a net force of zero—an

upward force opposite to the force of gravity.The upward force that balances the weight of an object on a surface is called the support force. A support force is often called the normal forceThe table pushes up on the book with as much force as the downward weight of the book.

• The upward support force is positive and the downward weight is negative.• The two forces add mathematically to zero. • Another way to say the net force on the book is zero is

F = 0.• The book lying on the table compresses atoms in the table and they squeeze upward on the book. The

compressed atoms produce the support force.The upward support force is as much as the downward pull of gravity.think!Suppose you stand on two bathroom scales with your weight evenly distributed between the two scales. What is the reading on each of the scales? What happens when you stand with more of your weight on one foot than the other?Answer: In the first case, the reading on each scale is half your weight. In the second case, if you lean more on one scale than the other, more than half your weight will be read on that scale but less than half on the other. The total support force adds up to your weight.Equilibrium of Moving ObjectsThe state of rest is only one form of equilibrium. An object moving at constant speed in a straight-line path is also in a state of equilibrium. Once in motion, if there is no net force to change the state of motion, it is in equilibrium.An object under the influence of only one force cannot be in equilibrium. Only when there is no force at all, or when two or more forces combine to zero, can an object be in equilibrium. When the push on the desk is the same as the force of friction between the desk and the floor, the net force is zero and the desk slides at an unchanging speed.If the desk moves steadily at constant speed, without change in its motion, it is in equilibrium.

• Friction is a contact force between objects that slide or tend to slide against each other. • In this case, F = 0 means that the force of friction is equal in magnitude and opposite in direction to the

pushing force.think!An airplane flies horizontally at constant speed in a straight-line direction. Its state of motion is unchanging. In other words, it is in equilibrium. Two horizontal forces act on the plane. One is the thrust of the propeller that pulls it forward. The other is the force of air resistance (air friction) that acts in the opposite direction. Which force is greater? Answer: Neither, for both forces have the same strength. Call the thrust positive. Then the air resistance is negative. Since the plane is in equilibrium, the two forces combine to equal zero.VectorsThe sum of two or more vectors is called their resultant. Combining vectors is quite simple when they are parallel:

• If they are in the same direction, they add. • If they are in opposite directions, they subtract.

a. The tension in the rope is 300 N, equal to Nellie’s weight. b. The tension in each rope is now 150 N, half of Nellie’s weight. In each case, F = 0.The Parallelogram RuleTo find the resultant of nonparallel vectors, we use the parallelogram rule. Consider two vectors at right angles to each other, as shown below. The constructed parallelogram in this special case is a rectangle. The diagonal is the resultant R.

In the special case of two perpendicular vectors that are equal in magnitude, the parallelogram is a square. The resultant is times one of the vectors. For example, the resultant of two equal vectors of magnitude 100 acting at a right angle to each other is 141.4.

When Nellie is suspended at rest from the two non-vertical ropes, is the rope tension greater or less than the tension in two vertical ropes? You need to use the parallelogram rule to determine the tension. Notice how the tension vectors form a parallelogram in which the resultant R is vertical.Nellie’s weight is shown by the downward vertical vector. An equal and opposite vector is needed for equilibrium, shown by the dashed vector. Note that the dashed vector is the diagonal of the parallelogram defined by the dotted lines. Using the parallelogram rule, we find that the tension in each rope is more than half her weight.

As the angle between the ropes increases, tension increases so that the resultant (dashed-line vector) remains at 300 N upward, which is required to support 300-N Nellie.

When the ropes supporting Nellie are at different angles to the vertical, the tensions in the two ropes are unequal.By the parallelogram rule, we see that the right rope bears most of the load and has the greater tension.

You can safely hang from a clothesline hanging vertically, but you will break the clothesline if it is strung horizontally.

think!Two sets of swings are shown at right. If the children on the swings are of equal weights, the ropes of which swing are more likely to break?Answer: The tension is greater in the ropes hanging at an angle. The angled ropes are more likely to break than the vertical ropes.think!Consider what would happen if you suspended a 10-N object midway along a very tight, horizontally stretched guitar string. Is it possible for the string to remain horizontal without a slight sag at the point of suspension?Answer: No way! If the 10-N load is to hang in equilibrium, there must be a supporting 10-N upward resultant. The tension in each half of the guitar string must form a parallelogram with a vertically upward 10-N resultant.

Newton’s Fist Law of MotionA ball at rest in the middle of a flat field is in equilibrium. No net force acts on it. If you saw it begin to move across the ground, you’d look for forces that don’t balance to zero.We don’t believe that changes in motion occur without cause.

Aristotle on MotionNatural motion on Earth was thought to be either straight up or straight down.

• Objects seek their natural resting places: boulders on the ground and smoke high in the air like the clouds.

• Heavy things fall and very light things rise. • Circular motion was natural for the heavens.

These motions were considered natural–not caused by forces.Violent motion, on the other hand, was imposed motion.

• It was the result of forces that pushed or pulled. • The important thing about defining violent motion was that it had an external cause. • Violent motion was imparted to objects.• Objects in their natural resting places could not move by themselves.

It was commonly thought for nearly 2000 years that a force was responsible for an object moving “against its nature.” • The state of objects was one of rest unless they were being pushed or pulled or moving toward their

natural resting place. • Most thinkers before the 1500s considered it obvious that Earth must be in its natural resting place. • A force large enough to move it was unthinkable. • Earth did not move.

Copernicus and the Moving EarthThe astronomer Nicolaus Copernicus (1473–1543) formulated a theory of the moving Earth.This idea was extremely controversial at the time. People preferred to believe that Earth was at the center of the universe.Copernicus worked on his ideas in secret. The first copy of his work, De Revolutionibus, reached him on the day of his death, May 24, 1543. Nicolaus Copernicus proposed that Earth moved around the sun. Galileo on MotionGalileo, the foremost scientist of late-Renaissance Italy, was outspoken in his support of Copernicus. One of Galileo’s great contributions to physics was demolishing the notion that a force is necessary to keep an object moving.Friction is the name given to the force that acts between materials that touch as they move past each other.

• Friction is caused by the irregularities in the surfaces of objects that are touching. • Even very smooth surfaces have microscopic irregularities that obstruct motion. • If friction were absent, a moving object would need no force whatever to remain in motion.

Galileo tested his idea by rolling balls along plane surfaces tilted at different angles. • A ball rolling down an inclined plane speeds up.• A ball rolling up an inclined plane—in a direction opposed by gravity—slows down.• A ball rolling on a smooth horizontal plane has almost constant velocity.

Galileo stated that if friction were entirely absent, a ball moving horizontally would move forever. No push or pull would be required to keep it moving once it is set in motion.Galileo’s conclusion was supported by another line of reasoning.

• He described two inclined planes facing each other.• A ball released to roll down one plane would roll up the other to reach nearly the same height. • The ball tended to attain the same height, even when the second plane was longer and inclined at a

smaller angle than the first plane. a. The ball rolling down the incline rolls up the opposite incline and reaches its initial height. b. The ball rolls a greater distance to reach its initial height. c. If there is no friction, the ball will never stop.

If the angle of incline of the second plane were reduced to zero so that the plane was perfectly horizontal, only friction would keep it from rolling forever. It was not the nature of the ball to come to rest as Aristotle had claimed. Galileo stated that this tendency of a moving body to keep moving is natural and that every material object resists changes to its state of motion. The property of a body to resist changes to its state of motion is called inertia.think!A ball is rolled across a counter top and rolls slowly to a stop. How would Aristotle interpret this behavior? How would Galileo interpret it? How would you interpret it? Answer: Aristotle would probably say that the ball stops because it seeks its natural state of rest. Galileo would probably say that the friction between the ball and the table overcomes the ball’s natural tendency to continue rolling—overcomes the ball’s inertia—and brings it to a stop. Only you can answer the last question! Newton’s Law of InertiaNewton’s first law, usually called the law of inertia, is a restatement of Galileo’s idea that a force is not needed to keep an object moving. Simply put, things tend to keep on doing what they’re already doing.

• Objects in a state of rest tend to remain at rest. • Only a force will change that state.

Now consider an object in motion. • In the absence of forces, a moving object tends to move in a straight line indefinitely. • Toss an object from a space station located in the vacuum of outer space, and the object will move

forever due to inertia.Blasts of air from many tiny holes provide a nearly friction-free surface on the air table. If you slide a hockey puck along the surface of a city street, the puck soon comes to rest. If you slide it along an air table where friction is practically absent, it slides with no apparent loss in speed. The law of inertia provides a completely different way of viewing motion from the ancients.

• Objects continue to move by themselves. • Forces are needed to overcome any friction that may be present and to set objects in motion initially. • Once the object is moving in a force-free environment, it will move in a straight line indefinitely.

think!A force of gravity between the sun and its planets holds the planets in orbit around the sun. If that force of gravity suddenly disappeared, in what kind of path would the planets move?Answer: Each planet would move in a straight line at constant speed.think!Is it correct to say that the reason an object resists change and persists in its state of motion is that it has inertia?Answer: We don’t know the reason why objects persist in their motion when nothing acts on them, but we know that they do, and we call this property inertia. Mass-A Measure of InertiaThe more mass an object has, the greater its inertia and the more force it takes to change its state of motion. The amount of inertia an object has depends on its mass—which is roughly the amount of material present in the object.Mass is a measure of the inertia of an object.You can tell how much matter is in a can when you kick it. Kick an empty can and it moves. Kick a can filled with sand and it doesn’t move as much.Do not confuse mass and volume.

• Volume is a measure of space and is measured in units such as cubic centimeters, cubic meters, and liters.

• Mass is measured in the fundamental unit of kilograms. Which has more mass, a feather pillow or a common automobile battery?Clearly an automobile battery is more difficult to set into motion. This is evidence of the battery’s greater inertia and hence its greater mass.The pillow has a larger size (volume) but a smaller mass than the battery. Mass is often confused with weight.

• We often determine the amount of matter in an object by measuring its gravitational attraction to Earth. However, mass is more fundamental than weight.

• Mass is a measure of the amount of material in an object. Weight, on the other hand, is a measure of the gravitational force acting on the object.

The amount of material in a particular stone is the same whether the stone is located on Earth, on the moon, or in outer space.

• The mass of the stone is the same in all of these locations. • The weight of the stone would be very different on Earth and on the moon, and still different in outer

space.The stone’s inertia, or mass, is a property of the stone and not its location.

The same force would be required to shake the stone with the same rhythm whether the stone was on Earth, on the moon, or in a force-free region of outer space. It’s just as difficult to shake a stone in its weightless state in space as it is in its weighted state on Earth.We can define mass and weight as follows:

• Mass is the quantity of matter in an object. More specifically, mass is a measure of the inertia, or “laziness,” that an object exhibits in response to any effort made to start it, stop it, or otherwise change its state of motion.

• Weight is the force of gravity on an object.Mass and weight are proportional to each other in a given place:

• In the same location, twice the mass weighs twice as much. • Mass and weight are proportional to each other, but they are not equal to each other.

It is common to describe the amount of matter in an object by its gravitational pull to Earth, that is, by its weight.• In the United States, the traditional unit of weight is the pound. In most parts of the world, however, the

measure of matter is commonly expressed in units of mass, the kilogram (kg).• At Earth’s surface, 1 kilogram has a weight of 2.2 pounds.

The SI unit of force is the newton. The SI symbol for the newton is N.One newton is equal to slightly less than a quarter pound. If you know the mass of something in kilograms and want its weight in newtons at Earth’s surface, multiply the number of kilograms by 10. One kilogram of nails weighs 10 newtons, which is equal to 2.2 pounds. Away from Earth’s surface, where the force of gravity is less, the bag of nails weighs less.think!Does a 2-kilogram bunch of bananas have twice as much inertia as a 1-kilogram loaf of bread? Twice as much mass? Twice as much volume? Twice as much weight, when weighed in the same location? Answer: Two kilograms of anything has twice the inertia and twice the mass of one kilogram of anything else. In the same location, where mass and weight are proportional, two kilograms of anything will weigh twice as much as one kilogram of anything. Except for volume, the answer to all the questions is yes. Bananas are much more dense than bread, so two kilograms of bananas must occupy less volume than one kilogram of bread. The Moving EarthCopernicus announced the idea of a moving Earth in the sixteenth century. One of the arguments against a moving Earth was:

• Consider a bird sitting at rest in the top of a tall tree.• The bird sees a worm, drops down vertically, and catches it. • It was argued that this would not be possible if Earth moved as Copernicus suggested. • The fact that birds do catch worms from high tree branches seemed to be clear evidence that Earth

must be at rest. You can refute this argument using the idea of inertia. Earth moves at 30 km/s, but so do the tree, the worm below, and even the air in between. Objects on Earth move with Earth as Earth moves around the sun.Earth does not need to be at rest for the bird to catch the worm. If we flip a coin in a high-speed car, bus, or plane, we can catch the vertically moving coin as we would if the vehicle were at rest. We see evidence for the law of inertia when the horizontal motion of the coin before, during, and after the catch is the same. The vertical force of gravity affects only the vertical motion of the coin. Flip a coin in an airplane, and it behaves as if the plane were at rest. The coin keeps up with you—inertia in action! Linear MotionEven things that appear to be at rest move. When we describe the motion of one object with respect to another, we say that the object is moving relative to the other object.

• A book that is at rest, relative to the table it lies on, is moving at about 30 kilometers per second relative to the sun.

• The book moves even faster relative to the center of our galaxy. The racing cars in the Indy 500 move relative to the track.When we discuss the motion of something, we describe its motion relative to something else.

• The space shuttle moves at 8 kilometers per second relative to Earth below. • A racing car in the Indy 500 reaches a speed of 300 kilometers per hour relative to the track. • Unless stated otherwise, the speeds of things in our environment are measured relative to the surface

of Earth.Although you may be at rest relative to Earth’s surface, you’re moving about 100,000 km/h relative to the sun.think!

A hungry mosquito sees you resting in a hammock in a 3-meters-per-second breeze. How fast and in what direction should the mosquito fly in order to hover above you for lunch?Answer: The mosquito should fly toward you into the breeze. When above you it should fly at 3 meters per second in order to hover at rest above you. SpeedBefore the time of Galileo, people described moving things as simply “slow” or “fast.” Such descriptions were vague. Galileo is credited as being the first to measure speed by considering the distance covered and the time it takes. Speed is how fast an object is moving.

Any combination of units for distance and time that are useful and convenient are legitimate for describing speed:• miles per hour (mi/h)• kilometers per hour (km/h)• centimeters per day• light-years per century

A cheetah is the fastest land animal over distances less than 500 meters and can achieve peak speeds of 100 km/h.We will primarily use the unit meters per second (m/s) for speed. If a cheetah covers 50 meters in a time of 2 seconds, its speed is 25 m/s.A car does not always move at the same speed. You can tell the speed of the car at any instant by looking at the car’s speedometer.The speed at any instant is called the instantaneous speed. The speedometer gives readings of instantaneous speed in both mi/h and km/h. In a trip by car, the car will certainly not travel at the same speed all during the trip. The driver cares about the average speed for the trip as a whole. The average speed is the total distance covered divided by the time.Average speed can be calculated easily:

For example, a distance of 240 kilometers during a time of 4 hours is an average speed of 60 km/h:

The average speed is often quite different from the instantaneous speed. Whether we talk about average speed or instantaneous speed, we are talking about the rates at which distance is traveled.If we know average speed and travel time, the distance traveled is easy to find. total distance covered = average speed × travel timeFor example, if your average speed is 80 kilometers per hour on a 4-hour trip, then you cover a total distance of 320 kilometers.think!If a cheetah can maintain a constant speed of 25 m/s, it will cover 25 meters every second. At this rate, how far will it travel in 10 seconds? In 1 minute?Answer: In 10 s the cheetah will cover 250 m, and in 1 min (or 60 s) it will cover 1500 m. think!The speedometer in every car also has an odometer that records the distance traveled. If the odometer reads zero at the beginning of a trip and 35 km a half hour later, what is the average speed?Answer:

In physics, velocity is speed in a given direction. • When we say a car travels at 60 km/h, we are specifying its speed. • When we say a car moves at 60 km/h to the north, we are specifying its velocity.

A quantity such as velocity that specifies direction as well as magnitude is called a vector quantity. • Speed is a scalar quantity. • Velocity, like force, is a vector quantity.

Constant speed means steady speed. Something with constant speed doesn’t speed up or slow down. Constant velocity means both constant speed and constant direction. Constant direction is a straight line, so constant velocity means motion in a straight line at constant speed. If either the speed or the direction (or both) is changing, then the velocity is changing.

• Constant speed and constant velocity are not the same. • A body may move at constant speed along a curved path but it does not move with constant velocity,

because its direction is changing every instant.The car on the circular track may have a constant speed but not a constant velocity, because its direction of motion is changing every instant.

think!The speedometer of a car moving northward reads 60 km/h. It passes another car that travels southward at 60 km/h. Do both cars have the same speed? Do they have the same velocity?Answer: Both cars have the same speed, but they have opposite velocities because they are moving in opposite directions.We can change the state of motion of an object by changing its speed, its direction of motion, or both. Acceleration is the rate at which the velocity is changing.

In physics, the term acceleration applies to decreases as well as increases in speed. The brakes of a car can produce large retarding accelerations, that is, they can produce a large decrease per second in the speed. This is often called deceleration. A car is accelerating whenever there is a change in its state of motion. Acceleration also applies to changes in direction.

• It is important to distinguish between speed and velocity. • Acceleration is defined as the rate of change in velocity, rather than speed. • Acceleration, like velocity, is a vector quantity because it is directional.

Accelerate in the direction of velocity–speed upAccelerate against velocity–slow downAccelerate at an angle to velocity–change directionWhen straight-line motion is considered, it is common to use speed and velocity interchangeably. When the direction is not changing, acceleration may be expressed as the rate at which speed changes.

Speed and velocity are measured in units of distance per time.• Acceleration is the change in velocity (or speed) per time interval.• Acceleration units are speed per time. • Changing speed, without changing direction, from 0 km/h to 10 km/h in 1 second, acceleration along a

straight line is

The acceleration is 10 km/h•s, which is read as “10 kilometers per hour-second.” Note that a unit for time appears twice: once for the unit of speed and again for the interval of time in which the speed is changing.think!Suppose a car moving in a straight line steadily increases its speed each second, first from 35 to 40 km/h, then from 40 to 45 km/h, then from 45 to 50 km/h. What is its acceleration? Answer: The speed increases by 5 km/h during each 1-s interval in a straight line. The acceleration is therefore 5 km/h•s during each interval.think!In 5 seconds a car moving in a straight line increases its speed from 50 km/h to 65 km/h, while a truck goes from rest to 15 km/h in a straight line. Which undergoes greater acceleration? What is the acceleration of each vehicle?Answer: The car and truck both increase their speed by 15 km/h during the same time interval, so their acceleration is the same. Imagine there is no air resistance and that gravity is the only thing affecting a falling object.

• An object moving under the influence of the gravitational force only is said to be in free fall. • The elapsed time is the time that has elapsed, or passed, since the beginning of any motion, in this case

the fall. During each second of fall the instantaneous speed of the object increases by an additional 10 meters per second. This gain in speed per second is the acceleration.

When the change in speed is in m/s and the time interval is in s, the acceleration is in m/s2, which is read as “meters per second squared.” The unit of time, the second, occurs twice—once for the unit of speed and again for the time interval during which the speed changes.For free fall, it is customary to use the letter g to represent the acceleration because the acceleration is due to gravity. Although g varies slightly in different parts of the world, its average value is nearly 10 m/s2. Where accuracy is important, the value of 9.8 m/s2 should be used for the acceleration during free fall. The instantaneous speed of an object falling from rest is equal to the acceleration multiplied by the elapsed time. v = gt

The letter v represents both speed and velocity. When the acceleration g = 10 m/s2 is multiplied by the elapsed time in seconds, the result is the instantaneous speed in meters per second.If a falling rock were somehow equipped with a speedometer, in each succeeding second of fall its reading would increase by the same amount, 10 m/s. The average speed of any object moving in a straight line with constant acceleration is the average of the initial speed and the final speed. The average speed of a freely falling object in its first second of fall is the sum of the initial speed of zero and the final speed of 10 m/s, divided by 2, or 5 m/s. Now consider an object thrown straight up:

• It moves upward for a while. • At the highest point, when the object is changing its direction from upward to downward, its

instantaneous speed is zero. • It then falls downward as if it had been dropped from rest at that height.

During the upward part of this motion, the object slows from its initial upward velocity to zero velocity. The object is accelerating because its velocity is changing. How much does its speed decrease each second? The speed decreases at the same rate it increases when moving downward—at 10 meters per second each second.

• The instantaneous speed at points of equal elevation in the path is the same whether the object is moving upward or downward.

• The velocities are different because they are in opposite directions. During each second, the speed or the velocity changes by 10 m/s downward.The change in speed each second is the same whether the ball is going upward or downward.think!During the span of the second time interval in Table 4.2, the object begins at 10 m/s and ends at 20 m/s. What is the average speed of the object during this 1-second interval? What is its acceleration?Answer: The average speed is 15 m/s. The acceleration is 10 m/s2. think!What would the speedometer reading on the falling rock be 4.5 seconds after it drops from rest? How about 8 seconds after it is dropped?Answer: The speedometer readings would be 45 m/s and 80 m/s, respectively. How far does an object in free fall travel in the first second?

• At the end of the first second, the falling object has an instantaneous speed of 10 m/s. • The initial speed is 0 m/s.• The average speed is 5 m/s. • During the first second, the object has an average speed of 5 m/s, so it falls a distance of 5 m.

Pretend that a falling rock is somehow equipped with an odometer. The readings of distance fallen increase with time.At the end of one second, the rock has fallen 5 meters. At the end of 2 seconds, it has dropped a total distance of 20 meters. At the end of 3 seconds, it has dropped 45 meters altogether.These distances form a mathematical pattern: at the end of time t, the object starting from rest falls a distance d.

We used freely falling objects to describe the relationship between distance traveled, acceleration, and velocity acquired. The same principles apply to any accelerating object. Whenever an object’s initial speed is zero and the acceleration a is constant, velocity and distance traveled are:

think!An apple drops from a tree and hits the ground in one second. What is its speed upon striking the ground? What is its average speed during the one second? How high above ground was the apple when it first dropped?

Answer: The speed when it strikes the ground is 10 m/s. The average speed was 5 m/s and the apple dropped from a height of 5 meters. Equations and tables are not the only way to describe relationships such as velocity and acceleration. Graphs can visually describe relationships.

On a speed-versus-time graph, the speed v of a freely falling object can be plotted on the vertical axis and time t on the horizontal axis.

• The “curve” that best fits the points forms a straight line. • For every increase of 1 s, there is the same 10 m/s increase in speed. • Mathematicians call this linearity. • Since the object is dropped from rest, the line starts at the origin, where both v and t are zero. • If we double t, we double v; if we triple t, we triple v; and so on. • This particular linearity is called a direct proportion, and we say that time and speed are directly proportional to

each other.The curve is a straight line, so its slope is constant. Slope is the vertical change divided by the horizontal change for any part of the line.For 10 m/s of vertical change there is a horizontal change of 1 s. The slope is 10 m/s divided by 1 s, or 10 m/s2. The straight line shows the acceleration is constant. If the acceleration were greater, the slope of the graph would be steeper.When the distance d traveled by a freely falling object is plotted on the vertical axis and time t on the horizontal axis, the result is a curved line. This distance-versus-time graph is parabolic.

The relationship between distance and time is nonlinear. The relationship is quadratic and the curve is parabolic—when we double t, we do not double d; we quadruple it. Distance depends on time squared!A curved line also has a slope—different at different points. The slope of a curve changes from one point to the next. The slope of the curve on a distance-versus-time graph is speed, the rate at which distance is covered per unit of time.The slope steepens (becomes greater) as time passes, which shows that speed increases as time passes. Air resistance noticeably slows the motion of things with large surface areas like falling feathers or pieces of paper. But air resistance less noticeably affects the motion of more compact objects like stones and baseballs. Drop a feather and a coin and the coin reaches the floor far ahead of the feather. Air resistance is responsible for these different accelerations. In a vacuum, the feather and coin fall side by side with the same acceleration, g.A feather and a coin accelerate equally when there is no air around them. In many cases the effect of air resistance is small enough to be neglected. With negligible air resistance, falling objects can be considered to be falling freely.Don’t mix up “how fast” with “how far.”

• How fast something freely falls from rest after a certain elapsed time is speed or velocity. The appropriate equation is v = gt.

• How far that object has fallen is distance. The appropriate equation is d = 1/2gt2. One of the most confusing concepts encountered in this subject is acceleration, or “how quickly does speed or velocity change.” What makes acceleration so complex is that it is a rate of a rate. It is often confused with velocity, which is itself a rate (the rate at which distance is covered). Acceleration is not velocity, nor is it even a change in velocity.

Projectile MotionIn the previous chapter we studied simple straight-line motion—linear motion. Now we extend these ideas to nonlinear motion—motion along a curved path. Throw a baseball and the path it follows is a combination of constant-velocity horizontal motion and accelerated vertical motion. A quantity that requires both magnitude and direction for a complete description is a vector quantity. Velocity is a vector quantity, as is acceleration.Other quantities, such as momentum, are also vector quantities.A quantity that is completely described by magnitude is a scalar quantity. Scalars can be added, subtracted, multiplied, and divided like ordinary numbers.

• When 3 kg of sand is added to 1 kg of cement, the resulting mixture has a mass of 4 kg. • When 5 liters of water are poured from a pail that has 8 liters of water in it, the resulting volume is 3

liters. • If a scheduled 60-minute trip has a 15-minute delay, the trip takes 75 minutes.

The airplane’s velocity relative to the ground depends on the airplane’s velocity relative to the air and on the wind’s velocity. The velocity of something is often the result of combining two or more other velocities.

• If a small airplane is flying north at 80 km/h relative to the surrounding air and a tailwind blows north at a velocity of 20 km/h, the plane travels 100 kilometers in one hour relative to the ground below.

• What if the plane flies into the wind rather than with the wind? The velocity vectors are now in opposite directions.

The resulting speed of the airplane is 60 km/h.Now consider an 80-km/h airplane flying north caught in a strong crosswind of 60 km/h blowing from west to east. The plane’s speed relative to the ground can be found by adding the two vectors.The result of adding these two vectors, called the resultant, is the diagonal of the rectangle described by the two vectors.An 80-km/h airplane flying in a 60-km/h crosswind has a resultant speed of 100 km/h relative to the ground.

think!Suppose that an airplane normally flying at 80 km/h encounters wind at a right angle to its forward motion—a crosswind. Will the airplane fly faster or slower than 80 km/h?Answer: A crosswind would increase the speed of the airplane and blow it off course by a predictable amount. Often we will need to change a single vector into an equivalent set of two component vectors at right angles to each other:

• Any vector can be “resolved” into two component vectors at right angles to each other. • Two vectors at right angles that add up to a given vector are known as the components of the given

vector. • The process of determining the components of a vector is called resolution.

A ball’s velocity can be resolved into horizontal and vertical components

Vectors X and Y are the horizontal and vertical components of a vector V.The horizontal component of motion for a projectile is just like the horizontal motion of a ball rolling freely along a level surface without friction. The vertical component of a projectile’s velocity is like the motion for a freely falling object. A projectile is any object that moves through the air or space, acted on only by gravity (and air resistance, if any). A cannonball shot from a cannon, a stone thrown into the air, a ball rolling off the edge of a table, a spacecraft circling Earth—all of these are examples of projectiles.Projectiles near the surface of Earth follow a curved path that at first seems rather complicated. These paths are surprisingly simple when we look at the horizontal and vertical components of motion separately.

Projectile motion can be separated into components. a. Roll a ball along a horizontal surface, and its velocity is constant because no component of gravitational force

acts horizontally. b. Drop it, and it accelerates downward and covers a greater vertical distance each second.

Most important, the horizontal component of motion for a projectile is completely independent of the vertical component of motion. Each component is independent of the other. Their combined effects produce the variety of curved paths that projectiles follow.There are two important things to notice in the photo of two balls falling simultaneously:

• The ball’s horizontal component of motion remains constant. Gravity acts only downward, so the only acceleration of the ball is downward.

• Both balls fall the same vertical distance in the same time. The vertical distance fallen has nothing to do with the horizontal component of motion.

The ball moves the same horizontal distance in the equal time intervals because no horizontal component of force is acting on it. The path traced by a projectile accelerating in the vertical direction while moving at constant horizontal velocity is a parabola. When air resistance is small enough to neglect, the curved paths are parabolic.think!At the instant a horizontally pointed cannon is fired, a cannonball held at the cannon’s side is released and drops to the ground. Which cannonball strikes the ground first, the one fired from the cannon or the one dropped?Answer: Both cannonballs fall the same vertical distance with the same acceleration g and therefore strike the ground at the same time. No matter the angle at which a projectile is launched, the vertical distance of fall beneath the idealized straight-line path is the same for equal times. This vertical distance is independent of what’s happening horizontally. With no gravity the projectile would follow the straight-line path (dashed line). But because of gravity it falls beneath this line the same vertical distance it would fall if it were released from rest. If there were no gravity the cannonball would follow the straight-line path shown by the dashed line. The vertical distance it falls beneath any point on the dashed line is the same vertical distance it would fall if it were dropped from rest:

For the component vectors of the cannonball’s motion, the horizontal component is always the same and only the vertical component changes. At the top of the path the vertical component shrinks to zero, so the velocity there is the same as the horizontal component of velocity at all other points. Everywhere else the magnitude of velocity is greater, just as the diagonal of a rectangle is greater than either of its sides.The velocity of a projectile is shown at various points along its path. Notice that the vertical component changes while the horizontal component does not. Air resistance is neglected.

The angle at which the projectile is launched affects the distance that it travels. Both projectiles have the same launching speed. The initial velocity vector has a greater vertical component than when the projection angle is less. This greater component results in a higher path. The horizontal component is less, so the range is less.Projectiles that are launched at the same speed but at different angles reach different heights (altitude) above the ground. They also travel different horizontal distances, that is, they have different horizontal ranges.

The paths of projectiles launched at the same speed but at different angles. The paths neglect air resistance.

The same range is obtained for two different projection angles—angles that add up to 90°.An object thrown into the air at an angle of 60° will have the same range as at 30° with the same speed. Maximum range is usually attained at an angle of 45°.Without air resistance, a projectile will reach maximum height in the same time it takes to fall from that height to the ground. The deceleration due to gravity going up is the same as the acceleration due to gravity coming down. The projectile hits the ground with the same speed it had when it was projected upward from the ground.Without air resistance, the speed lost while the cannonball is going up equals the speed gained while it is coming down. The time to go up equals the time to come down.In the presence of air resistance, the path of a high-speed projectile falls below the idealized parabola and follows the solid curve. think!A projectile is launched at an angle into the air. Neglecting air resistance, what is its vertical acceleration? Its horizontal acceleration?Answer: Its vertical acceleration is g because the force of gravity is downward. Its horizontal acceleration is zero because no horizontal force acts on it. think!At what point in its path does a projectile have minimum speed?Answer: The minimum speed of a projectile occurs at the top of its path. If it is launched vertically, its speed at the top is zero. If it is projected at an angle, the vertical component of velocity is still zero at the top, leaving only the horizontal component. Newton’s Second Law of MotionThe cause of acceleration is force.When a hockey puck is at rest, the net force on it (gravity and the support force) is balanced, so the puck is in equilibrium. Hit the puck (that is, apply an unbalanced force to it) and the puck experiences a change in motion—it accelerates. Apply another force by striking the puck again, and the puck’s motion changes again.Recall from the previous chapter that the combination of forces acting on an object is the net force.

• Acceleration depends on the net force. • To increase the acceleration of an object, you must increase the net force acting on it. • An object’s acceleration is directly proportional to the net force acting on it:

acceleration ~ net force(The symbol ~ stands for “is directly proportional to.”)

Push on an empty shopping cart. Then push equally hard on a heavily loaded shopping cart.The loaded shopping cart will accelerate much less than the empty cart. Acceleration depends on the mass being pushed.The same force applied to twice as much mass results in only half the acceleration. The acceleration is inversely proportional to the mass.

Inversely means that the two values change in opposite directions. As the denominator increases, the whole quantity decreases by the same factor.The acceleration produced depends on the mass that is pushed.Newton’s second law states that the acceleration produced by a net force on an object is directly proportional to the magnitude of the net force, is in the same direction as the net force, and is inversely proportional to the mass of the object.

Newton’s second law describes the relationship among an object's mass, an object's acceleration, and the net force on an object.

By using consistent units, such as newtons (N) for force, kilograms (kg) for mass, and meters per second squared (m/s2) for acceleration, we get the exact equation:

If a is acceleration, F is net force, and m is mass,

The acceleration is equal to the net force divided by the mass. • If the net force acting on an object doubles, its acceleration is doubled.• If the mass is doubled, then acceleration will be halved. • If both the net force and the mass are doubled, the acceleration will be unchanged.

think!If a car can accelerate at 2 m/s2, what acceleration can it attain if it is towing another car of equal mass? Answer: The same force on twice the mass produces half the acceleration, or 1 m/s2. do the math!A car has a mass of 1000 kg. What is the acceleration produced by a force of 2000 N?

do the math!If the force is 4000 N, what is the acceleration?

Doubling the force on the same mass simply doubles the acceleration.do the math!How much force, or thrust, must a 30,000-kg jet plane develop to achieve an acceleration of 1.5 m/s2? Arrange Newton’s second law to read:

force = mass × accelerationF = ma = (30,000 kg)(1.5 m/s2) = 45,000 kg•m/s2

= 45,000 NFriction is a force and affects motion:

• Friction acts on materials that are in contact with each other.• It always acts in a direction to oppose relative motion. • When two solid objects come into contact, the friction is mainly due to irregularities in the two surfaces.

Rubber against concrete produces more friction than steel against steel, so concrete road dividers have replaced steel rails. The friction produced by a tire rubbing against the concrete is more effective in slowing the car than the friction produced by a steel car body sliding against a steel rail. A concrete road divider has a better design than a steel road divider for slowing an out-of-control, sideswiping car.The concrete divider is wider at the bottom to ensure that the tire will make contact with the divider before the steel car body does.Both liquids and gases are called fluids because they flow.

• Fluid friction occurs as an object pushes aside the fluid it is moving through. • The friction of liquids is appreciable, even at low speeds. • Air resistance is the friction acting on something moving through air.

When friction is present, an object may move with a constant velocity even when an outside force is applied to it. In such a case, the friction force just balances the applied force. The net force is zero, so there is no acceleration.A diagram showing all the forces acting on an object is called a free-body diagram.The direction of the force of friction always opposes the direction of motion. a. Push the crate to the right and friction acts toward the left.b. The sack falls downward and air friction acts upward.

think!Two forces act on a book resting on a table: its weight and the support force from the table. Does a force of friction act as well? Answer: No, not unless the book tends to slide or does slide across the table. Friction forces occur only when an object tends to slide or is sliding. The amount of force per unit of area is called pressure. When the force is perpendicular to the surface area,

P is the pressure and A is the area over which the force acts.Pressure is measured in newtons per square meter, or pascals (Pa). One newton per square meter is equal to one pascal.The force of the book on the table is the same.The upright book exerts the same force, but greater pressure, against the supporting surface. You exert more pressure against the ground when you stand on one foot than when you stand on both feet due to the decreased area of contact. The smaller the area supporting a given force, the greater the pressure on that surface.think!In attempting to do the bed-of-nails demonstration, would it be wise to begin with a few nails and work upward to more nails?Answer: No, no, no! There would be one less physics teacher if the demonstration were performed with fewer nails. The resulting greater pressure would cause harm. Galileo showed that falling objects accelerate equally, regardless of their masses.

• This is strictly true if air resistance is negligible, that is, if the objects are in free fall. • It is approximately true when air resistance is very small compared with the mass of the falling object.

In Galileo’s famous demonstration, a 10-kg cannonball and a 1-kg stone strike the ground at practically the same time.This experiment demolished the Aristotelian idea that an object that weighs ten times as much as another should fall ten times faster than the lighter object. Recall that mass (a quantity of matter) and weight (the force due to gravity) are proportional.

• A 10-kg cannonball experiences 10 times as much gravitational force (weight) as a 1-kg stone. • Newton’s second law tells us to consider the mass as well. • Ten times as much force acting on ten times as much mass produces the same acceleration.

F stands for the force (weight) acting on the cannonball, and m stands for the correspondingly large mass of the cannonball. The small F and m stand for the weight and mass of the stone.

• The ratio of weight to mass is the same for these or any objects. • All freely falling objects undergo the same acceleration at the same place on Earth.

The ratio of weight (F) to mass (m) is the same for the 10-kg cannonball and the 1-kg stone.The weight of a 1-kg stone is 10 N at Earth’s surface. Using Newton’s second law, the acceleration of the stone is

The weight of a 10-kg cannonball is 100 N at Earth’s surface and the acceleration of the cannonball is

A feather and a coin fall with equal accelerations in a vacuum, but very unequally in the presence of air. When falling in air, the coin falls quickly while the feather flutters to the ground. The force due to air resistance diminishes the net force acting on the falling objects.You experience the force due to air resistance when you stick your hand out of the window of a moving car.

• If the car moves faster, the force on your hand increases. • If instead of just your hand, you hold your physics book out the window with the large side facing

forward, the air resistance force is much larger than on your hand at the same speed. Air resistance force ~ speed × frontal areaThe expression shows that the air resistance force is directly proportional to the speed and frontal area of an object.Terminal speed is the speed at which the acceleration of a falling object is zero because friction balances the weight. Terminal velocity is terminal speed together with the direction of motion.Sky divers reach terminal speed when air resistance equals weight.A falling feather reaches its terminal speed quite quickly. Its area is large relative to its very small weight so air resistance has a large effect on the feather’s motion. A coin has a relatively small area compared to its weight, so the coin will have to fall faster to reach its terminal speed.

The terminal speed for a sky diver varies from about 150 to 200 km/h, depending on the weight and orientation of the body.

• A heavier person will attain a greater terminal speed than a lighter person. • Body orientation also makes a difference. More air is encountered when the body is spread out and

surface area is increased.The flying squirrel increases its area by spreading out. This increases air resistance and decreases the speed of its fall.Terminal speed can be controlled by variations in body orientation.

• A heavy sky diver and a light sky diver can remain in close proximity to each other if the heavy person spreads out like a flying squirrel while the light person falls head or feet first.

• A parachute greatly increases air resistance, and cuts the terminal speed down to 15 to 25 km/h, slow enough for a safe landing.

At low speeds, air resistance is often negligible, but at high speeds, it can make quite a difference. If you hold a baseball and tennis ball at arm’s length and release them at the same time, you’ll see them strike the floor at the same time. But if you drop them from the top of a building, you’ll notice the heavier baseball strikes the ground first. think!Which experiences a greater air resistance force, a falling piece of paper or a falling elephant?Answer: The elephant! It has a greater frontal area and falls faster than a piece of paper—both of which mean the elephant pushes more air molecules out of the way. The effect of the air resistance force on each, however, is another story! think!If a heavy person and a light person open their parachutes together at the same altitude and each wears the same size parachute, who will reach the ground first?Answer: The heavy person will reach the ground first. Like a feather, the light person reaches terminal speed sooner, while the heavy person continues to accelerate until a greater terminal speed is reached.