Force & Momentum

Force is defined as the rate of change of momentum

$F = \frac{∆ p}{∆ t}$

Where:
- F = force in newtons (N)
- p = momentum in kilogram metres per second (kg m s^-1)
- t = time in seconds (s)
- Δ (the Greek letter delta) = change in

Change in momentum, Δp, can also be expressed as:

change in momentum = final momentum − initial momentum

$∆ p = p_{f i n a l} - p_{i n i t a l} = p_{f} - p_{i}$

Force and momentum are vector quantities
- They can have a positive or negative direction

Worked example

A car of mass 1500 kg hits a wall at an initial velocity of 15 m s^-1.

It then rebounds off the wall at 5 m s^-1 and comes to rest after 3.0 s.

Calculate the average force experienced by the car.

Answer:

Step 1: List the known quantities and assign direction:

Mass, m = 1500 kg
Inital velocity, vi = 15 m s^-1
Final velocity, v_f = −5 m s^-1
Time taken, t = 3.0 s

Step 2: State the equation for force and momentum

$F = \frac{∆ p}{∆ t}$

Step 3: State the equation for change in momentum

$∆ p = p_{f} - p_{i}$

Step 4: Calculate the initial momentum

$p_{i} = m \times v_{i}$

$p_{i} = 1500 \times 15$

$p_{i} = 22 500 kg m s^{- 1}$

Step 5: Calculate the final momentum

$p_{f} = m \times v_{f}$

$p_{f} = 1500 \times - 5$

$p_{f} = - 7500 kg m s^{- 1}$

Step 6: Calculate the change in momentum

$∆ p = p_{f} - p_{i}$

$∆ p = - 7500 - 22 500$

$∆ p = - 30 000 kg m s^{- 1}$

Step 7: Calculate the force

$F = \frac{∆ p}{∆ t}$

$F = \frac{- 30 000}{3}$

$F = - 10 000 N$

Examiner Tip

It is easy to forget to convert values to SI units and to assign directions to values during the calculation because your brain is focused on so many other things. Unfortunately, students often lose marks for this in exams. Therefore, it is good practice to get into the habit of performing these steps before you begin the calculation.

Direction of forces

The force that is equal to the rate of change of momentum is still the resultant force
The force on an object will be negative if the direction of the force opposes the direction of its initial velocity
This means that a force is exerted by the object it has collided with

Forces acting between a car and a wall upon impact

Direction of forces, downloadable AS & A Level Physics revision notes

The force exerted by the wall on car will be equal in magnitude and opposite in direction to the force exerted by the car on the wall: F_car = –F_wall

A car collides with a wall, and the car exerts a force of 300 N on the wall
The wall also exerts a force of −300 N on the car
The force exerted by the wall on the car is equal in magnitude and opposite in direction to the force exerted by the car on the wall
This is Newton’s third law of motion (see Newton’s laws of motion)

Time of impact

The force exerted is also determined by the time taken for the impact to occur
The same change in momentum, over a longer period of time will exert less force, and vice versa

$F = \frac{∆ p}{∆ t}$

- As Δt increases, F decreases, when Δp remains the same
- As Δt decreases, F increases, when Δp remains the same

Worked example

A tennis ball hits a racket with a change in momentum of 0.5 kg m s^-1.

The first ball has a contact time of 2.0 s. The second ball has a contact time of 0.1 s.

Determine which ball exerts the greatest force on the racket.

Answer:

Step 1: List the known quantities

Change in momentum, Δp = 0.5 kg m s^-1
Change in time for the first ball, Δt₁ = 2.0 s
Change in time for the second ball, Δt₂ = 0.1 s

Step 2: Determine the force exerted by the first ball on the racket

$F = \frac{∆ p}{∆ t_{1}}$

$F = \frac{0.5}{2.0}$

$F = 0.25 N$

Step 3: Determine the force exerted by the second ball on the racket

$F = \frac{∆ p}{∆ t_{2}}$

$F = \frac{0.5}{0.1}$

$F = 5.0 N$

Step 4: State which ball exerted the greatest force on the racket

The second tennis ball exerted the greatest force on the racket

Examiner Tip

In exam questions, look carefully at the cause and effect - which object is exerting a force on the other. Look out for key terms such as 'acts upon' and 'exerts'. It can be helpful to sketch a quick diagram to help you visualise the situation. This is particularly important when multiple values are given in the question, because not all of them may be relevant to the calculation.

Force & Momentum (CIE AS Physics)

Revision Note

Force & momentum

Worked example

Examiner Tip

Direction of forces

Forces acting between a car and a wall upon impact

Time of impact

Worked example

Examiner Tip

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Author: Leander