Lesson 3 of 4Energy and Momentum

Unit 2 · Lesson 3 of 4

Momentum and Impulse

Momentum as mass times velocity, the impulse that changes it, reading impulse as the area under a force-time graph, and why helmets, airbags and crumple zones work by stretching out the time of a collision.

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Key things to remember

1Momentum: $p = m v$ , a vector in $kg \cdot m/s$ that points along the velocity.
2Impulse-momentum theorem: $F_{n e t} Δ t = Δ p = m v_{2} - m v_{1}$ .
3Impulse is the area under a force-time graph; $1 N \cdot s = 1 kg \cdot m/s$ .
4Average force: $F_{a vg} = Δ p /Δ t$ . For the same $Δ p$ , a longer time means a smaller force.
5A rebound reverses the velocity, so its change in momentum is larger than for stopping alone.

A baseball catcher's mitt is padded, and a catcher lets their hands move back as the ball arrives. A gymnast lands on a thick mat and bends their knees. A car's front end is built to crumple. All three are solving the same problem: an object has to be stopped, and the question is how to stop it without an enormous force. The quantity that has to be removed is momentum, and the tool that removes it is impulse.

Momentum

p = m v

Momentum is mass times velocity, measured in kilogram metres per second.

Momentum is a vector: it points the same way as the velocity, so in one dimension you must choose a positive direction and give it a sign. Its unit is $kg \cdot m/s$ . It measures how hard an object is to stop. A slow truck and a fast bullet can both be hard to stop, because momentum grows with mass and with speed. Unlike kinetic energy, $\frac{1}{2} m v^{2}$ , which is a scalar and never negative, two objects moving in opposite directions can have momenta that add to zero.

Impulse changes momentum

F_{n e t} Δ t = Δ p = m v_{2} - m v_{1}

The impulse-momentum theorem. Impulse is measured in newton seconds, the same unit as momentum.

This is Newton's second law written a different way. Substitute $a = (v_{2} - v_{1}) /Δ t$ into $F_{n e t} = m a$ and multiply both sides by $Δ t$ . The product $F_{n e t} Δ t$ is the impulse, and it equals the change in momentum. Signs matter: a ball that hits a wall at $10 m/s$ and bounces back at $8 m/s$ changes its velocity by $18 m/s$ , not $2 m/s$ , because it has to stop and then move the other way. A bounce always needs a larger impulse than simply stopping.

Fig 1A large force for a short time and a small force for a long time can give the same area, and so the same impulse.

Real collision forces are not constant: they rise quickly and fall away again. The impulse is still well defined, because it is the area under the force-time graph. You can then find the average force, the constant force that would give the same area over the same time: $F_{a vg} = Δ p /Δ t$ . For a fixed change in momentum, the area is fixed, so a longer contact time must mean a lower average force, and a lower peak force too.

Designing for safety

Fig 2A crumple zone stretches the stopping time, so the same change in momentum needs a much smaller force.

Technology	What it does	The physics
Crumple zone	The front and rear of a car fold up in a controlled way	Increases the stopping time of the passenger compartment, lowering the force; the crushing metal also absorbs kinetic energy
Airbag	Inflates in front of the occupant, then vents as they press into it	Increases stopping time and spreads the force over a larger area of the body
Seat belt	Holds the occupant to the car, and stretches slightly	Makes the occupant slow down with the car over its crumple time instead of stopping suddenly on the dashboard
Helmet	A hard shell over a crushable foam liner	The liner crushes, increasing the stopping time of the head; the shell spreads the force over a wider area
Child car seat	A harness and padded shell sized to the child	Keeps a light child moving with the seat and spreads the stopping force over the strongest parts of the body
Gym mats, bent knees	Soft surfaces and flexing joints	Stretch the landing over a longer time, so the force on bones and joints is smaller

The rule behind every safety device

In a crash the change in momentum is set by the mass and the speed; nothing can change it. What a design can change is the time over which it happens. Because

F_{a vg} = Δ p /Δ t

, making the stopping time

10

times longer makes the average force

10

times smaller.

Trade-offs in design

Safety design is never just physics. A helmet must protect but stay light enough to wear; a stiffer car body protects occupants in some crashes but can transfer more force to them in others, so engineers balance a rigid passenger cage against crushable ends. Crash tests with instrumented dummies measure forces and accelerations on the body so these trade-offs can be tested before people are exposed to them.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

0.145 kg

baseball arrives at a bat at

38 m/s

and leaves in the opposite direction at

45 m/s

. The contact lasts

1.0 ms

. Find the impulse on the ball and the average force of the bat.

Example 2

70 kg

car passenger moving at

15 m/s

is brought to rest in a collision. Compare the average force on the passenger if an airbag stops them in

0.10 s

with the force if they hit the dashboard and stop in

0.010 s

Example 3

0.45 kg

soccer ball at rest is kicked. The force rises steadily from

0

to a peak of

2000 N

and falls steadily back to

0

, over a total contact time of

0.010 s

. Find the impulse and the ball's speed as it leaves the foot.

Example 4

1500 kg

car travels at

25 m/s

[E]. Its brakes supply a net force of

6000 N

[W] for

3.0 s

. Find its final velocity.

Where marks get lost

Ignoring direction when an object bounces. A ball hitting a wall at $10 m/s$ and rebounding at $8 m/s$ has $Δ v = 18 m/s$ , not $2 m/s$ .
Saying an airbag reduces the change in momentum. The momentum change is the same; the airbag increases the time and so reduces the force.
Using the peak force from a force-time graph as if it acted the whole time. Impulse is the area under the curve, which for a triangle is half of peak times duration.
Leaving the contact time in milliseconds. $1.5 ms$ is $1.5 \times 1 0^{- 3} s$ ; using $1.5$ gives a force a thousand times too small.
Treating momentum as a scalar like kinetic energy. Momentum has direction, and in one dimension that means a sign.

Flashcards 7

Momentum

Impulse

Impulse-momentum theorem

Impulse on a force-time graph

+ 3 more

Practice

What is the magnitude of the momentum of a

0.060 kg

tennis ball moving at

50 m/s

, in

kg \cdot m/s

kg·m/s

Which has more momentum: a

2000 kg

car rolling at

5.0 m/s

, or a

10 g

bullet moving at

400 m/s

A constant force of

250 N

acts on a hockey puck for

0.020 s

. What impulse does it deliver, in

N \cdot s

N·s

0.50 kg

ball moving at

6.0 m/s

is caught and stopped in

0.015 s

. What is the size of the average force on it, in newtons?

0.20 kg

ball hits a wall at

10 m/s

and rebounds straight back at

8.0 m/s

. What is the magnitude of its change in momentum, in

kg \cdot m/s

kg·m/s

Take right as positive. A

0.40 kg

cart moving at

2.0 m/s

to the right receives an impulse of

1.2 N \cdot s

to the left. What is its final velocity, in m/s, including the sign?

m/s

Two identical eggs fall from the same height, one onto a pillow and one onto a tile floor. Which quantity is the same for both eggs as they are stopped?

How does an airbag reduce injuries in a collision?

Momentum is measured in

kg \cdot m/s

. What equivalent unit, built from newtons, is usually used for impulse?

1.0 kg

object moving at

10 m/s

is brought to rest in different ways. Order them from the largest average force to the smallest.

1Stopped by a wooden board in 0.01 s
2Stopped by a long net in 1 s
3Stopped by a foam pad in 0.1 s
4Stopped by concrete in 0.001 s

Match each safety feature to how it reduces the force on a person.

Crumple zone

Helmet foam liner

Seat belt

Bending your knees on landing

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