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TRANSCRIPT
Describing Motion: Kinematics
in One Dimension
Chapter 2
Definitions
Kinematics – branch of Physics that
describes how objects move
Translational motion – when an object
moves without rotating
Scalars vs. Vectors
Scalar – quantity that represents magnitude only (with units)
Magnitude – numerical quantity
Vector – quantity that represents magnitude (with units) and direction
Use arrows to represent
Length of arrow is proportional to magnitude
Tip of arrow indicates direction
VERY helpful to draw vector diagrams to work problems
Often have arrow drawn on top of symbol to be distinguished from scalars
Vectors can be positive or negative.
Think of a Cartesian plane.
X axis - Designate right as positive and
left as negative.
Y axis - Designate up as positive and
down as negative.
Vectors can be added graphically
Vectors must have the same units to do this.
Connect the tail of the first vector to the head of the last vector. This line is called the resultant.
For one dimensional motion, the resultant’s direction is the same as the vector with the largest magnitude.
To subtract vectors, add the opposite of the vector.
Distance and Displacement
Distance – total path covered by the
object from start to end (scalar)
Displacement – shortest path from start
to end (vector)
Which statement is true regarding
distance and displacement? A. Distance will never be greater than
displacement.
B. Displacement will never be greater than
distance.
C. Distance and displacement will always be
equal.
D. Distance and displacement will never be equal.
Displacement
Displacement
= change in position (x )
= final position – initial position (xf – xi)
x = xf – xi (horizontal motion)
y = yf – yi (vertical motion)
Example 1
A space shuttle takes off from FL and circles the Earth several times, finally landing in CA. While the shuttle is in flight, a photographer flies from FL to CA to take pictures of the astronauts as the step off the shuttle. Who has the greater displacement – the astronauts or photographer?
a. The astronauts
b. The photographer
c. They both have the same displacement.
d. There is not enough info to determine the answer.
Example 2
What is the coach’s distance travelled?
a. 0 yds e. -5 yds
b. 5 yds f. -55 yds
c. 55 yds g. -95 yds
d. 95 yds
Example 3
What is the skier’s displacement?
a. 40 m right e. 40 m left
b. 80 m right f. 80 m left
c. 100 m right g.100 m left
d. 140 m right h. 140 m left
Example 4
You start walking home from school. After walking
1.3 km North, you get a phone call on your cell
from your mom asking if you can meet her at
the mall. You will have to turn around and
walk 2.5 km South. Determine your distance
to get to the mall.
a. 1.2 km e. 1.2 km N i. 2.5 km N
b. 1.3 km f. 1.2 km S j. 2.5 km S
c. 2.5 km g. 1.3 km N k. 3.8 km N
d. 3.8 km h. 1.3 km S l. 3.8 km S
Example 4
You start walking home from school. After walking
1.3 km North, you get a phone call on your cell
from your mom asking if you can meet her at
the mall. You will have to turn around and
walk 2.5 km South. Determine your
displacement to get to the mall.
a. 1.2 km e. 1.2 km N i. 2.5 km N
b. 1.3 km f. 1.2 km S j. 2.5 km S
c. 2.5 km g. 1.3 km N k. 3.8 km N
d. 3.8 km h. 1.3 km S l. 3.8 km S
Speed
Scalar quantity that represents how fast
an object is moving
Rate at which an object covers a
distance (vavg = dtotal / ttotal)
Instantaneous speed measured by your
speedometer
Velocity
Vector quantity that represents the rate
at which an object changes position
(displacement / time)
vavg = x / t = xf-xi / tf-ti
SI Units = m/s and must include direction!
Example 1
Freddie walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. The entire motion lasted for 24 seconds. Determine the average speed.
a. 0 m/s e. 0.2 m/s W
b. 0.3 m/s f. 0.3 m/s N
c. 0.5 m/s g. 0.5 m/s E
d. 2 m/s h. 1 m/s S
Example 2
Freddie walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. The entire motion lasted for 24 seconds. Determine the average velocity.
a. 0 m/s e. 0.2 m/s W
b. 0.3 m/s f. 0.3 m/s N
c. 0.5 m/s g. 0.5 m/s E
d. 2 m/s h. 1 m/s S
Example 3
A car drives along the highway at 115
km/h for 2.50 h. Once in the city, the
car drives at 60.0 km/h for the next
0.500 h. Determine the average speed
of the car.
a. 58.3 km/h
b. 87.5 km/h
c. 106 km/h
Determining Velocity Graphically
Velocity can be determined from a
position-time or d-t graphs
Slope = y coordinates / x coordinates
= displacement /time
= m/s
= VELOCITY!
d-t graphs
+ y axis = positive direction
- y axis = negative direction
The x-axis represents your starting point.
Linear Slopes on d-t graphs
Horizontal line = displacement is zero (stationary)
Positive slope = moving at a constant velocity away
from start (+y-axis) or moving at a constant velocity
back toward start (-y-axis) = constant positive
velocity!
Negative slope = moving at a constant velocity
toward start (+y-axis) or moving at a constant velocity
away from starting point (-y-axis) = constant negative
velocity!
Example of a d-t graph
d-t graphs
Nonlinear graphs
When a graph is curved, the slope at each
point is the slope of a tangent line
Since each point has a different tangent,
curved d-t graphs show changing velocity
Non-linear Slopes on d-t graphs
A curve means the displacement is changing
at a non-constant rate (variable velocity)
If the steepness of the slope (tangent line)
increases, velocity is increasing in magnitude.
If the steepness of the slope (tangent line)
decreases, velocity is decreasing in
magnitude.
Nonlinear slope d-t graphs
Graphing Example
Describe the motion:
a. Stationary
b. Moving away from start at a constant velocity
c. Moving toward start at a constant velocity
d. Increasing velocity
e. Decreasing velocity
A
B C
D
E
F
Motion Diagram
Like a strobe photograph that shows
slow motion of object
Show displacement of object over equal
time intervals
If you walked in snow or mud…
What would the spacing of your footprints
look like if you were walking…
1. At a slow, constant pace
2. Then you started to speed up
3. Then slowed down until
4. Finally you stop?
Motion Map Parts
The dot: Indicates the position of the
object
The arrow (vector): indicates the
direction and speed of the object.
Dots and Arrows Together
Dot alone = not moving.
Dot and arrow together:
Position, direction and speed.
Direction and Size
Right = positive
direction
Left = negative
direction
The longer the
arrow, the greater
the velocity.
SLOW
FASTER
FASTEST
The grid….
Motion Maps are drawn along a grid to show
the position of the object.
Draw a minimum of 3 arrows to show a
pattern.
Forward, Constant Velocity, Slow
Forward, Constant Velocity, Faster
Motion Maps
Suppose that you took a stroboscopic picture of a car
moving to the right at constant velocity where each image
revealed the position of the car at one-second intervals.
What would it look like if the car were moving faster?
Series of Motion
•The object moves forward at constant velocity,
•then stops and remains in place for two seconds,
•then moves backward at a slower constant velocity.
Example 1
Draw a motion map.
The car is accelerating!
Each successive arrow
is longer, indicating the
velocity is increasing.
Example 2
Follow the arrows to
describe the motion…
Given the following motion diagram,
describe the object’s motion:
x0 x1 x2 x3 x4
Indicate position arrows (vectors)
1. The object is not moving.
2. The object is moving at a constant velocity.
3. The object is increasing speed.
4. The object is decreasing speed.
Sue runs towards Jim starting from rest. The dots in the motion
diagram below represent Sue’s position at 0.2-second time
intervals. Which diagram could represent Sue's initial motion?
Which motion diagram could represent Sue’s motion once she has
reached her maximum speed?
Acceleration
Vector quantity that represents a change in
VELOCITY of an object
Includes a change in speed or direction (or both)
May be positive or negative
Positive when
Increasing speed and traveling right or up (+/+)
Decreasing speed and traveling left or down (-/-)
Negative when
Decreasing speed and traveling right or up (-/+)
Increasing speed and traveling left or down (+/-)
a = v = vf – vi
t t
Units are in m/s2
When acceleration is constant,
Changes in position (velocity);
changes in velocity (acceleration)
x4-x3 (x1-x0)/t~v0 (x2-x1)/t~v1 x3-x2
In equal time intervals (t)
(visualizing acceleration can be difficult)
v1-v0 v2-v1 v3-v2
Velocity is shown by the arrows connecting two successive dots.
The following drawings indicate the motion of a ball from left to
right . Each circle represents the position of the ball at succeeding
instants of time. Each time interval between successive positions
is equal.
Rank each case from the highest to the lowest acceleration based
on the ball's motion using the coordinate system specified by the
dashed arrows in the figures. Note: Zero is greater than negative,
and ties are possible.
Acceleration Due to Gravity
Often called free fall acceleration
Because any object on Earth is so small compared to the size of Earth, we have a constant value for acceleration close to the Earth’s surface.
g = -10 m/s2: The value is negative because it is ALWAYS in the downward direction!
Used for value of a whenever there is vertical motion (up or down).
Free Fall Motion Diagram
Acceleration is more than just a formula –
Applying definitions, we get useful relationships
• vavg= x/t
• xf = xi + vavg t
• a=v/t
• vf = vi + a t
• If a is constant, then vavg = vf + vi
2
One Dimensional Motion Equations
Using the definitions of average velocity and
acceleration, we can derive 4 mathematical
equations for motion in one dimension.
x = ½ (vf + vi) t vf = vi + a t
x = vi t + ½ at2 vf2 = vi
2 + 2 ax
*Note: these formulas can only be used when an object
has a constant acceleration.
*For vertical motion, replace x with y and a with g
Determining Acceleration Graphically
Acceleration can be determined from a
velocity-time or v-t graph
Slope = y coordinates / x coordinates
= velocity /time
= m/s2
= ACCELERATION!
v-t graphs
+ y axis = positive direction
- y axis = negative direction
Crossing the x-axis means you changed
directions. It does not represent your
starting point!
Linear Slopes on v-t graphs
Horizontal line = constant velocity
Positive slope = increasing speed at a constant rate
in the positive direction or decreasing speed at a
constant rate in the negative direction (constant +
acceleration)
Negative slope = decreasing speed at a constant rate
in the positive direction or increasing speed at a
constant rate in the negative direction(constant –
acceleration)
Non-linear Slopes on v-t graphs
A curve means the velocity is changing
at a non-constant rate (variable
acceleration)
If the steepness of the slope increases,
acceleration is increasing in magnitude.
If the steepness of the slope decreases,
acceleration is decreasing in magnitude.
v-t graph example
What time interval(s) are in the
positive direction?
1. 0-3 s 4. 0-8 s
2. 0-5.5 s 5. 13-16.5 s
3. 0-5.5 s, 13-16.5 s
What time interval(s) are in the
negative direction?
1. 5.5 - 6.5 s 4. 5.5 – 13 s
2. 5.5 – 8 s 5. 8-13 s
3. 5.5 – 9 s 6. 8 – 16.5 s
What time interval(s) show positive
acceleration?
1. 0 - 3 s 4. 0 – 3, 13 – 16.5 s
2. 0 – 5.5 s 5. 9 - 16.5 s
3. 0 – 8 s 6. 13 – 16.5 s
What time interval(s) show negative
acceleration?
1. 3 – 5.5 s 4. 8 – 16.5 s
2. 5.5 - 8 s 5. 8 – 9, 13 - 16.5 s
3. 5.5 - 9 s
How to determine the distance and
displacement from a v-t graph
v t = x (so on this graph, that is y x, or length x width)
To find the area under the curve, separate into triangles and/or rectangles.
Calculate the area of each object and add them all together.
For distance, add all magnitudes together ignoring + or –
For displacement, you must consider the sign as + or – and then add together.
Area under the curve example
Find the displacement during these
55 seconds of motion