4th form: Motion & Forces

Scalars and Vectors

• Measurable quantities can be divided into scalars and vectors.

• Scalar quantities have a magnitude (size) only– Examples?

• Mass, distance, speed, time...

• Vector quantities have both a magnitude and a direction associated with them– Examples?

• Force, displacement, velocity...

Adding scalars and vectors

• Adding scalar quantities is easy, just add the magnitudes– eg if it takes 15 minutes to eat lunch and you

have a further 45 minutes before lessons, how long was the break?

• Adding vectors, we have to take the direction of the quantities into account– eg pushing a car against friction

Adding vectors

Adding vectors (more generally)

• The resultant R is found by combining all the component vectors together– It is the single vector which is equivalent to

the action of all the component vectors

• This is A-level

stuff...

Speed: a reminder• Speed is a measure of how quickly

something is moving

• Actually, the above formula really tells you the average speed during the time interval

• As the time interval gets smaller, you get closer to calculating the instantaneous speed.

takentime travelleddistance

speed

Displacement-time graphs• Try to describe the motion shown in the graph

– What does the slope of the line represent?– What does the slope of the dotted line tell you?

Displacement-time graphsConstant speed forward

stationary

Constant speed backwards

After 160 minutes, we are back where we started

Slope=average speed of return journey

Slope = speed

Speed=5/0.42=11.9 km/h

Calculating speed• The slope of the graph gives the speed

(strictly the velocity)–

• The steeper the line, the higher the speed

takentime travelleddistance

slope

Slope = 60/10 = 6.0 m/s

(a)

(b)

(c)

(a)

(c)

(d)

Slope = -100/25 = -4.0 m/s

Slope = 40/15 = 2.7 m/s

(d)

Slope = 0/5 = 0.0 m/s(b)

Displacement-time graphs

Note: distance can also become negative, if object travels in the opposite direction

How would you represent something getting slower?

t

x

Speed and Velocity

• The velocity of an object gives its instantaneous speed and direction– (This is called a VECTOR)

• As with displacement, the sign of the velocity indicates the direction– a negative velocity means speed in the

opposite direction

Speed and Velocity

• Going from A to B: + velocity

• Going from C to F: - velocity

Velocity-time graphs• Try to describe the motion shown in the graph

– What does the slope of the line represent?– Where is the object not moving?

Velocity-time graphsConstant acceleration Constant speed

forwards Gradual slowing

More rapid slowing

stationary

Reversing direction and speeding up

Constant speed backwards

Slowing to a stop

Acceleration

• Acceleration is the rate of change of velocity

– If you are speeding up, acceleration is +– If you are slowing down, acceleration is -

takentime

yin velocit changeonaccelerati

m/s2

m/s

s

Acceleration is the slope of the velocity graph

Acceleration = (v-u)/t

So for this region:

a= 8/4 = 2 m/s2

and for this region:

a= 0/6 = 0 m/s2 (constant v)

• Displacement = velocity × time• i.e. the area under the graph

So in the first 4s:

In the next 6 seconds

What about the displacement?

m 16

482

1 travelleddistance

m 64 s 10in distance totalso

m 84

86 travelleddistance

Tachographs

• A tachograph is an instrument which records the velocity-time graph of a vehicle.

• It is used to check that EU regulations limiting the time lorry and bus drivers can spend at the wheel are obeyed– 9 hours/day– 45 minute break every 4.5

hrs.

Forces: a reminder• A force is a “push” or a “pull”. Unit: newton (N)

• Forces arise due to the interaction of two (or more) objects.

• Not all forces require contact, some can act at a distance– e.g. gravity, magnetism

• Forces are vectors,

Direction matters

Weight: a reminder

• “Mass” is a measure of how much “stuff” an object contains– measured in kg

• “Weight” is the force that object exerts due to the effect of gravity– measured in newtons

(N)

• So an astronaut has the same mass on Earth or the Moon, but his weight will be different

mgW weight mass

gravitational field strength

On Earth, g ≈10 N/kg

Representing forces

• Forces can be represented with arrows, whose length indicates the size of the force.

Force diagrams

• A free body diagram can be very useful to analyse the forces acting on an object

• We draw it isolated from its surroundings and show all the forces acting

What forces are acting here?• Draw on as many as you can think of…

Tension in the rope

Weight and reaction(for each person)

Push and friction(for each person)

Combining forces

• If several forces act on an object, we can work out the equivalent single resultant force by adding them up, taking direction into account.

• What is the resultant?– 4 newtons downwards

• How about these?

7 N

6 N6 N

3 N

4 N

0 N

4 N 4 N4 N

1 N5 N

Balanced forces

• It is possible to have all forces balanced, so the resultant = 0.

• In this case, no resultant force acts and the object continues to move at constant velocity (or remain stationary if it wasn’t moving).

Newton’s 1st law

For a plane flying at constant speed and height:

Thrust = drag

Lift = weight

Newton’s 1st Law

• A body will remain at rest or, if moving, continue to move at a constant velocity, unless acted on by a force.

Unbalanced forces

• If the resultant force is not zero, a net force is acting on the body and its motion will change.

• It will accelerate in the direction of the force.

• thrust > drag and lift > weight,so aeroplane accelerates and takes off

Force causes acceleration

• When a force acts on a body, it changes its velocity

• If no resultant force acts, there is no acceleration (Newton’s 1st law)

• Remember, acceleration can mean a change of speed or direction

• 1 N is the force which accelerates 1 kg at 1 m/s2

Newton’s 2nd law

amF force mass acceleration

F=ma

• So:– For a given mass, a

bigger force produces a bigger acceleration

– For a given force, a smaller mass experiences a bigger acceleration

Force, mass and acceleration1) A force of 1000 N is applied to push a mass of 500 kg.

How quickly does it accelerate?

2) A force of 3000N acts on a car to make it accelerate by1.5 m/s2. How heavy is the car?

3) A car accelerates at a rate of 5 m/s2. If it weighs 500 kg how much driving force is the engine applying?

4) A force of 10 N is applied by a boy while lifting a 20 kg mass. How much does it accelerate by?

Remember Weight?

• We hadwhere W was the weight – the force due to gravity• Now we know

mgW

Acceleration due to gravity

gravitational field strength

On Earth: g ≈10 N/kg, a ≈10 m/s2

ga

mgmaWF

maF

or

and so

Investigating F, m and a

• We can measure acceleration with light gates

• What happens as you vary:– The mass on the

hanger?– The mass of the

trolley?

• Why do we need a ramp?

• How do we set the right angle?

What do we find?

• acceleration is proportional to force

• acceleration is inversely proportional to force

a F

a 1/m

Horizontal motion

• Driving force < counter force: vehicle slows down

• Driving force = counter force: vehicle moves at constant velocity

• Driving force > counter force: vehicle speeds up

Counter force

Driving force Driving force – provided by rider/engine

Counter force – air resistance and friction

Falling Objects

An object falls because of its weight (force due to gravity)

When object falls freely – no other forces act on it so resultant force is just its weight.

Remember F = ma?

Acceleration of 10m/s2 is constant for all objects.

• So if we dropped a hammer and a feather at the same time, which would hit the ground first? Why?

• Hammer & Feather

Classic experiment

Drag

• Objects moving in a fluid have drag force. • For objects travelling through the air we call

this drag force air resistance.

• Air resistance increases with speed.

• So as a falling object speeds up, the resultant force decreases. This means the acceleration decreases.

Reaching a constant velocity

Object reaches a constant velocity when the drag force/air resistance is equal & opposite to its weight.

Resultant force = zeroAcceleration = zeroVelocity = terminal velocity

Why does a car have a top speed?

The AR 8C has a 4.7 litre 450 bhp (340 kW) engine to provide driving force.Force means acceleration, so why can’t the car accelerate forever?

What determines terminal velocity?

• Frontal area• Shape• Mass• Surface

Stages of a parachute jump

Just after letting go...

•Velocity =0•Drag = 0•Force = weight•Acceleration = g

Falling quite fast now...

•Velocity is high•Drag is large•Force < weight•Acceleration < g

Falling at a constant speed...

•Velocity constant•Drag = weight•Force = 0•Acceleration = 0

Pull the ripcord...

•Velocity still high•Drag > weight•Force upwards•Acceleration upwards (so speed of fall decreases)

Drift downward...

•Velocity constant (slower)•Drag = weight•Force = 0•Acceleration = 0

Label the graph

Urban Myth?

• So, would a penny dropped from a skyscraper kill someone it hit at the bottom?

• See here for the answer

• (also here if you wonder

about bullets coming down)

Springs: a reminder

• We have seen that springs obey Hooke’s Law:– The extension is proportional to the force

applied (up to some limit)

kxF

Other “stretchy” things

• Hooke’s Law also applies to other objects...– Metal bars, wires, bones, even glass!

• ...up to a point– If you go beyond that point you may get failure

(snap) or permanent deformation (doesn’t return to original shape)

Hooke’s Law limit

Rubber bands

• Rubber bands are elastic, can be stretched and return to their original length, BUT they do not obey Hooke’s Law

• How can you tell?

• Describe how it stretches...