Lesson 4 of 4Energy and Momentum

Unit 2 · Lesson 4 of 4

Collisions in One and Two Dimensions

Conservation of momentum, telling elastic from inelastic collisions by checking kinetic energy, explosions and recoil, and solving two-dimensional collisions by components.

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Key things to remember

1Conservation of momentum for an isolated system: $m_{1} v_{1} + m_{2} v_{2} = m_{1} v_{1}^{'} + m_{2} v_{2}^{'} .$
2Momentum is conserved in every collision and explosion; kinetic energy is conserved only in elastic collisions.
3Perfectly inelastic (objects stick): $m_{1} v_{1} + m_{2} v_{2} = (m_{1} + m_{2}) v^{'} .$
4Elastic, head-on, target at rest: $v_{1}^{'} = \frac{m _{1} - m _{2}}{m _{1} + m _{2}} v_{1}$ and $v_{2}^{'} = \frac{2 m _{1}}{m _{1} + m _{2}} v_{1}$ .
5In two dimensions, conserve the $x$ and $y$ components of momentum separately.

During a collision the forces are huge, brief and almost impossible to measure: two billiard balls touch for a tiny fraction of a second. Yet you can predict exactly how they move afterward without knowing those forces at all. The reason is that one quantity, the total momentum of the objects, is the same after the collision as before. This topic uses that law, together with a check on kinetic energy, to analyze collisions and explosions in one and two dimensions.

Conservation of momentum

m_{1} v_{1} + m_{2} v_{2} = m_{1} v_{1}^{'} + m_{2} v_{2}^{'}

Total momentum before equals total momentum after, as long as no net external force acts on the system.

Here is why. During the collision, object 1 pushes on object 2 and, by Newton's third law, object 2 pushes back on object 1 with an equal and opposite force. The two forces last for exactly the same time, so the impulses are equal and opposite: whatever momentum object 2 gains, object 1 loses. The total cannot change. This holds for an isolated system, one with no net external force. Friction from the ground acts during a car crash, but for the short time of the impact it is small compared with the collision forces, so momentum is very nearly conserved.

Elastic and inelastic collisions

Type	Total momentum	Total kinetic energy	Example
Elastic	Conserved	Conserved	Colliding steel balls or air-track gliders with spring bumpers (very nearly)
Inelastic	Conserved	Decreases; some becomes thermal energy, sound and deformation	A basketball bouncing off another ball
Perfectly inelastic	Conserved	Decreases by the largest possible amount	Two cars that lock together; a dart sticking in a block
Explosion	Conserved	Increases; stored chemical or elastic energy is released	Fireworks; two carts pushed apart by a released spring

Momentum is conserved in every type. The type is decided by kinetic energy: calculate $\sum \frac{1}{2} m v^{2}$ before and after. If it is unchanged, the collision is elastic. If objects stick together, it is perfectly inelastic and a single final velocity replaces $v_{1}^{'}$ and $v_{2}^{'}$ . When a target of mass $m_{2}$ starts at rest and the collision is elastic and head-on, solving both conservation laws together gives the results below.

v_{1}^{'} = \frac{m _{1} - m _{2}}{m _{1} + m _{2}} v_{1} v_{2}^{'} = \frac{2 m _{1}}{m _{1} + m _{2}} v_{1}

One-dimensional elastic collision with object 2 initially at rest.

Three cases worth knowing

m_{1} = m_{2}

, then

v_{1}^{'} = 0

and

v_{2}^{'} = v_{1}

: the moving object stops dead and the target takes all its velocity, as in a billiard shot straight on. If

m_{1}

is much larger than

m_{2}

, the heavy object barely slows and the light target flies off at almost

2 v_{1}

. If

m_{1}

is much smaller,

v_{1}^{'}

is negative: the light object bounces back at almost its original speed.

Loading interactive…

Set the masses and velocities, choose elastic or perfectly inelastic, and compare momentum and kinetic energy before and after.

Explosions, recoil and collisions in two dimensions

In an explosion the objects start together, often at rest, so the total momentum is zero before and must be zero after: the pieces fly apart with momenta that cancel. A rifle recoils backward when it fires a bullet forward, and a rocket moves forward by pushing exhaust gas backward. Kinetic energy is not conserved, since it comes from stored chemical or elastic energy. In two dimensions momentum is still a vector, so it is conserved separately in each direction. Resolve every momentum into $x$ and $y$ components, write one conservation equation for each, then recombine the final components with the Pythagorean theorem and $tan θ$ .

Fig 1The momentum vectors after the collision add tip to tail to give the same total as before.

Predicting the neutrino

In beta decay a nucleus emits an electron. If only the electron and the nucleus were involved, conservation of energy and momentum would require every electron to leave with the same energy, and the electron and nucleus to recoil in exactly opposite directions. Experiments in the early twentieth century found instead that the electrons came out with a range of energies. Rather than give up the conservation laws, Wolfgang Pauli proposed in 1930 that an undetected neutral particle was carrying away the missing energy; such a particle would carry momentum too. It was named the neutrino, and it was detected directly in 1956. Canada's Sudbury Neutrino Observatory later helped show that neutrinos have mass.

Worked examples

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

Example 1

1200 kg

car travelling at

18 m/s

[E] rear-ends an

1800 kg

truck travelling at

8.0 m/s

[E]. They lock together. Find their velocity just after the collision and the kinetic energy lost.

Example 2

0.50 kg

cart moving at

2.0 m/s

to the right hits a

0.30 kg

cart at rest. Afterward the

0.50 kg

cart moves at

0.50 m/s

to the right. Find the velocity of the other cart and decide whether the collision is elastic.

Example 3

1.5 kg

cart and a

0.50 kg

cart sit at rest, pressed against a compressed spring between them. The spring is released and the

0.50 kg

cart moves off at

1.2 m/s

to the left. Find the velocity of the other cart and the energy the spring released.

Example 4

On an air table, a

0.20 kg

puck moving at

3.0 m/s

[E] collides with a

0.30 kg

puck moving at

1.5 m/s

[N]. They stick together. Find their velocity afterward.

Where marks get lost

Dropping the signs of velocities in opposite directions. Two carts moving toward each other must have momenta of opposite sign.
Assuming kinetic energy is conserved in every collision. Only momentum is always conserved; check kinetic energy before calling a collision elastic.
Using one mass instead of the combined mass for objects that stick together after a perfectly inelastic collision.
Adding the speeds of two-dimensional momenta directly. Resolve into components, conserve $x$ and $y$ separately, then recombine.
Thinking an explosion from rest has no momentum afterward because the pieces are moving. Each piece has momentum; the vector total is zero.

Flashcards 8

Law of conservation of momentum

Elastic collision

Inelastic collision

Perfectly inelastic collision

+ 4 more

Practice

3.0 kg

cart moving at

4.0 m/s

hits a

1.0 kg

cart at rest, and they stick together. What is their speed afterward, in m/s?

m/s

Take right as positive. A

2.0 kg

ball moving at

4.0 m/s

to the right collides head-on with a

3.0 kg

ball moving at

2.0 m/s

to the left. They stick. What is their final velocity, in m/s, with sign?

m/s

Which quantity is conserved in every collision between two objects with no net external force?

Two carts collide and bounce apart. The total kinetic energy afterward is less than before. What kind of collision is this?

4.0 kg

rifle at rest fires a

10 g

bullet at

400 m/s

. What is the rifle's recoil speed, in m/s?

m/s

A billiard ball rolling at speed

v

hits an identical ball at rest head-on in an elastic collision. What happens?

When a firework shell bursts, what happens to the total kinetic energy of the pieces compared with the shell just before it burst?

2.0 kg

cart moving at

6.0 m/s

collides with a

1.0 kg

cart at rest, and they stick. How much kinetic energy is lost, in joules?

0.50 kg

puck moving at

4.0 m/s

[E] strikes an identical puck at rest. Afterward the first puck moves at

2.0 m/s

6 0^{\circ}

N]. What is the speed of the second puck, in m/s? Give three significant figures.

m/s

Which particle was proposed in 1930 to explain the energy and momentum that seemed to go missing in beta decay?

Order the steps for solving a two-dimensional collision.

1Combine the components into a magnitude and direction
2Solve for the unknown final components
3Write conservation of momentum separately for x and for y
4Choose positive x and y directions
5Resolve every initial momentum into x and y components

Match each event to the type of collision or interaction.

Two air-track gliders bounce off spring bumpers with no kinetic energy lost

A dart sticks in a hanging wooden block

A tennis ball bounces off a racket, losing some kinetic energy

Two skaters at rest push apart

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