Impulse
 The force and momentum equation can be rearranged to find the impulse
 Impulse, I, is equal to the change in momentum:
I = FΔt = Δp = mv – mu
 Where:
 I = impulse (N s)
 F = force (N)
 t = time (s)
 p = momentum (kg m s^{–1})
 m = mass (kg)
 v = final velocity (m s^{–1})
 u = initial velocity (m s^{–1})
 This equation is only used when the force is constant
 Since the impulse is proportional to the force, it is also a vector
 The impulse is in the same direction as the force
 The unit of impulse is N s
 The impulse quantifies the effect of a force acting over a time interval
 This means a small force acting over a long time has the same effect as a large force acting over a short time
Examples of Impulse
 An example in everyday life of impulse is when standing under an umbrella when it is raining, compared to hail (frozen water droplets)
 When rain hits an umbrella, the water droplets tend to splatter and fall off it and there is only a very small change in momentum
 However, hailstones have a larger mass and tend to bounce back off the umbrella, creating a greater change in momentum
 Therefore, the impulse on an umbrella is greater in hail than in rain
 This means that more force is required to hold an umbrella upright in hail compared to rain
Since hailstones bounce back off an umbrella, compared to water droplets from rain, there is a greater impulse on an umbrella in hail than in rain
Worked Example
A 58 g tennis ball moving horizontally to the left at a speed of 30 m s^{–1} is struck by a tennis racket which returns the ball back to the right at 20 m s^{–1}.
(i) Calculate the impulse delivered to the ball by the racket
(ii) State which direction the impulse is in
(i)
Step 1: Write the known quantities

 Taking the initial direction of the ball as positive (the left)
 Initial velocity, u = 30 m s^{–1}
 Final velocity, v = –20 m s^{–1}
 Mass, m = 58 g = 58 × 10^{–3} kg
Step 2: Write down the impulse equation
Impulse I = Δp = m(v – u)
Step 3: Substitute in the values
I = (58 × 10^{–3}) × (–20 – 30) = –2.9 N s
(ii)
Direction of the impulse

 Since the impulse is negative, it must be in the opposite direction to which the tennis ball was initial traveling (since the left is taken as positive)
 Therefore, the direction of the impulse is to the right
Impulse on a ForceTime Graph
 In real life, forces are often not constant and will vary over time
 If the force is plotted against time, the impulse is equal to the area under the forcetime graph
When the force is not constant, the impulse is the area under a force–time graph
 This is because
Impulse = FΔt
 Where:
 F = force (N)
 Δt = change in time (s)
 The impulse is therefore equal whether there is
 A small force over a long period of time
 A large force over a small period of time
 The forcetime graph may be a curve or a straight line
 If the graph is a curve, the area can be found by counting the squares underneath
 If the graph is made up of straight lines, split the graph into sections. The total area is the sum of the areas of each section
Worked Example
A ball of mass 3.0 kg, initially at rest, is acted on by a force F which varies with t as shown by the graph.
Calculate the magnitude of the velocity of the ball after 16 s.
Step 1: List the known quantities

 Mass, m = 3.0 kg
 Initial velocity, u = 0 m s^{1} (since it is initially at rest)
Step 2: Calculate the impulse

 The impulse is the area under the graph
 The graph can be split up into two rightangled triangles with a base of 8 s and a height of 4 kN
Area = Impulse = 32 × 10^{3} N s
Step 3: Write the equation for impulse
Impulse, I = Δp = m(v – u)
Step 4: Substitute in the values
I = mv
32 × 10^{3} = 3.0 × v
v = (32 × 10^{3}) ÷ 3.0
v = 10666 m s^{–1} = 11 km s^{1}
Step 5: State the final answer

 The final magnitude of the velocity of the ball is:
v = 11 km s^{1}
Exam Tip
 Remember that if an object changes direction, then this must be reflected by the change in sign of the velocity. As long as the magnitude is correct, the final sign for the impulse doesn’t matter as long as it is consistent with which way you have considered positive (and negative)
 For example, if the left is taken as positive and therefore the right as negative, an impulse of 20 N s to the right is equal to 20 N s
Some maths tips for this section:
Rate of Change
 ‘Rate of change’ describes how one variable changes with respect to another
 In maths, how fast something changes with time is represented as dividing by Δt (e.g. acceleration is the rate of change in velocity)
 More specifically, Δt is used for finite and quantifiable changes such as the difference in time between two events
Areas
 The area under a graph may be split up into different shapes, so make sure you’re comfortable with calculating the area of squares, rectangles, rightangled triangles and trapeziums!
Impulse in Context
In Sports
 For example, in cricket:
 A cricket ball travels at very high speeds and therefore has a high momentum
 When a fielder catches the ball, it exerts a force onto their hands
 Stopping a ball with high momentum at once will cause a large force onto their hands
 This is because a change in momentum (impulse) acts over a short period of time which creates a large force on the fielder’s hands and could cause serious injury
 A fielder moves their hands back when they catch the ball, which increases the time for its change in momentum to reduce
 This means there will be less force exerted on the fielder’s hands and therefore less chance of injury
A cricket fielder moves their hands backwards when catching a cricket ball to reduce the force it will exert on their hands
 In football:

 Increasing the contact time is sometimes used to advantage, as the longer the contact time, the larger change in momentum
 When kicking a football, after a strong kick the motion is followed through
 This creates a large impulse and the ball then has a higher velocity
The follow through action of a football kick increases the change in momentum of the ball
Worked Example
A tennis ball hits a racket with a change in momentum of 0.5 kg m s^{1}.
For the different contact times, which tennis racket experiences more force from the tennis ball?
Momentum conservation and safety
 The force of an impact in a vehicle collision can be decreased by increasing the contact time over which the collision occurs
 The contact time is the time in which the vehicle or the passenger is in contact with what it has collided with
 Vehicles have safety features such as crumple zones, seat belts and airbags to account for this
 For a given force upon impact, these absorb the energy from the impact and increase the time over which the force takes place
 This, in turn, increases the time taken for the change in momentum of the passenger and the vehicle to come to rest
 The increased time reduces the force and risk of injury on a passenger
The seat belt, airbag and crumple zones help reduce the risk of injury on a passenger
Designing Safety Features
 Vehicle safety features are designed to absorb energy upon an impact by changing shape
 Seat belts
 These are designed to stop a passenger from colliding with the interior of a vehicle by keeping them fixed to their seat in an abrupt stop
 They are designed to stretch slightly to increase the time for the passenger’s momentum to reach zero and reduce the force on them in a collision
 Airbags
 These are deployed at the front on the dashboard and steering wheel when a collision occurs
 They act as a soft cushion to prevent injury on the passenger when they are thrown forward upon impact
 Crumple zones
 These are designed into the exterior of vehicles
 They are at the front and back and are designed to crush or crumple in a controlled way in a collision
 This is why vehicles after a collision look more heavily damaged than expected, even for relatively small collisions
 The crumple zones increase the time over which the vehicle comes to rest, lowering the impact force on the passengers
 The effect of the increase in time and force can be shown on a forcetime graph
 For the same change in momentum, which depends on the mass and speed of a vehicle, the increase in contact time will result in a decrease in the maximum force exerted on the vehicle and passenger
 This is demonstrated by a lower peak and wider base on a forcetime graph
The increase in contact time Δt decreases the force for the same impulse
Worked Example
A 7 kg bowling ball has an impulse of 84 Ns act upon it. The bowling ball was initially at rest and sitting on a flat frictionless surface. What is the distance moved by the bowling ball in the first 3 seconds after the impulse was delivered?
Step 1: List the known quantities

 Mass of the bowling ball: 7 kg
 Impulse acting on the bowling ball: 84 Ns
 Bowling ball initial velocity (at rest): 0 ms^{1}
 Time of movement: 3 s
Step 2: Find the velocity caused by the impulse

 The velocity caused by the impulse can be found from the equation linking mass, velocity, and impulse:
Impulse, I = Δp = m(v – u)
Step 3: Rearrange and solve for v
I = m × v (since u = 0ms^{1})
v = I ÷ m = 84 ÷ 7 = 12 ms^{1}
Step 4: Find the distance traveled

 This can be found using time and velocity
v = d ÷ t
d = v × t
d = 12 × 3 = 36 m
Step 5: State the final answer

 The bowling ball moved 36 m