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AP®PhysicsCollege BoardPhysics 1: Algebra-BasedStudy GuidesUnit 4: Linear MomentumConservation of Linear MomentumMomentum in Collisions

Momentum in Collisions (College Board AP® Physics 1: Algebra-Based)

Study Guide

Katie M

Written by: Katie M

Reviewed by: Caroline Carroll

Momentum in collisions

  • The conservation of momentum principle can be used to analyze collisions

  • A collision is defined as

An event in which two or more moving objects come together and exert a force on each another

  • In a collision, it is assumed that

    • the internal forces between objects are much larger than the net external force exerted on the system

    • the time interval during which the momentum changes is very short

Collisions in one dimension

  • Collisions in one dimension (1D) usually involve two objects interacting on a single straight line

    • This means the velocity vectors before and after the collision are directed along the same axis and can be either positive or negative

  • When solving problems involving collisions in one dimension:

    • identify the objects included in the system, making sure there are no net external forces

    • write expressions for the momentum of each object before and after the collision

    • write expressions for the total momentum of the system before and after the collision

    • apply the conservation of momentum principle

    • solve for the unknown quantity

A collision in one dimension between two objects A and B along the same line. Before the collision, object A moves towards stationary object B with velocity v0A. After the collision, A and B move away from each other with velocities vA, and vB respectively.
During a collision in one dimension between two objects, A and B, they move along the same line before and after the collision

  • Applying the conservation of momentum principle (defining the positive direction to the right):

m subscript A v with rightwards arrow on top subscript 0 A end subscript space equals space m subscript B v with rightwards arrow on top subscript B space minus space m subscript A v with rightwards arrow on top subscript A

  • Since the velocity of each object is directed along the same axis, the vector notation can be removed:

m subscript A over m subscript B space equals space fraction numerator v subscript B over denominator v subscript 0 A end subscript space plus space v subscript A end fraction

Collisions in two dimensions

  • Collisions in two dimensions (2D) usually involve an interaction between two objects in a plane

    • This means the velocity vectors before and after the collision may have both horizontal and vertical components

  • When solving problems involving collisions in two dimensions:

    • identify the objects included in the system, making sure there are no net external forces

    • write expressions for the x- and y- components of the momentum of each object before and after the collision

    • write expressions for the total momentum of the system in each direction before and after the collision

    • apply the conservation of momentum principle to each direction separately

    • solve for the unknown quantity using Pythagoras' theorem

A collision in two dimensions between two objects A and B. Before the collision, object A moves towards stationary object B with velocity v0A. After the collision, object A moves away at an angle θA above the horizontal with velocity vA and object B moves away at an angle θB below the horizontal with velocity vB.
During a collision in two dimensions between two objects, A and B, they move off in different directions after the collision

  • If no external forces act on a system, horizontal and vertical components of momentum are conserved

  • Applying the conservation of momentum principle:

m subscript A v with rightwards arrow on top subscript 0 A end subscript space equals space m subscript A v with rightwards arrow on top subscript A space plus space m subscript B v with rightwards arrow on top subscript B

  • The components of the velocities in the x-direction are:

m subscript A v subscript 0 A end subscript space equals space m subscript A v subscript A space cos space theta subscript A space plus space m subscript B v subscript B space cos space theta subscript B

  • The components of the velocities in the y-direction are:

0 space equals space m subscript A v subscript A space sin space theta subscript A space minus space m subscript B v subscript B space sin space theta subscript B

For a collision in two dimensions between two objects A and B, the velocities must be resolved into x- and y- components. Object A moves away at an angle θA above the horizontal with velocity vA, so it has a velocity of vAcosθA in the x-direction and vAsinθA in the y-direction. Object B moves away at an angle θB below the horizontal with velocity vB, so it has a velocity of vBcosθB in the x-direction and vBsinθBin the y-direction.
To solve problems involving collisions in two dimensions, velocity vectors must be resolved into their horizontal and vertical components

Worked Example

A particle moves to the right and collides with a stationary particle. The direction of the second particle after the collision is shown below.

Two particles labeled 1 and 2 before and after a collision. Before the collision, particle 1 moves right with velocity v1, particle 2 is stationary. After the collision, particle 2 changes direction and particle 1's direction is unknown.

Which of the following is a possible direction of the first particle after the collision?

Four arrows labeled A, B, C, and D. A points up-left, B points left, C points down-right, and D points right.

The correct answer is C

Answer:

Step 1: Analyze the scenario

  • Before the collision, the total momentum is directed horizontally, as momentum is in the same direction as the velocity

  • The total momentum of the system must be conserved

  • Therefore, the net momentum after the collision must also be horizontal

Step 2: Apply the principle of conservation of momentum

  • Since particle 2 is moving up after the collision, particle 1 must move down at the same angle to cancel out the vertical component of the momentum

  • This is shown by option C

Vector components are shown: a leftward arrow splitting into two angled vectors. Vertical components cancel, horizontal components combine.

Examiner Tips and Tricks

In AP Physics 1 you will be expected to tackle conservation of momentum problems in one dimension both quantitatively and qualitatively, whereas for problems in two dimensions, you will only be expected to tackle these qualitatively and semiquantitatively.

Exam questions will predominantly be focused on your ability to set up the equations properly and reason about how changing a given mass, speed, or angle would affect other quantities.

In AP Physics 2, you will be expected to solve conservation of momentum problems in two dimensions that include one unknown final velocity, so it's worth getting to grips with the mathematics involved now.

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    Katie M

    Author: Katie M

    Expertise: Physics

    Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.

    Caroline Carroll

    Author: Caroline Carroll

    Expertise: Physics Subject Lead

    Caroline graduated from the University of Nottingham with a degree in Chemistry and Molecular Physics. She spent several years working as an Industrial Chemist in the automotive industry before retraining to teach. Caroline has over 12 years of experience teaching GCSE and A-level chemistry and physics. She is passionate about creating high-quality resources to help students achieve their full potential.