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Elastic collisions review

Review the key concepts, equations, and skills for elastic collisions, including how to predict objects' final velocities.

Key terms

Term (symbol)Meaning
Closed systemSystem that is not acted on by a net external force. Also called an isolated system.
Conservation of momentumIn a closed (isolated) system, momentum is constant.
Elastic collisionCollision where both momentum and kinetic energy are conserved. There is no change in kinetic energy in the system as a result of the collision.

Equations

EquationSymbolsMeaning in words
pi=pfpi and pf are the total initial and final momentumThe total initial momentum equals the total final momentum for a closed system. Commonly called the conservation of momentum.

How to predict final velocities for an elastic collision

We know a collision is elastic if kinetic energy is conserved:
12m1v1i2+12m2v2i2=12m1v1f2+12m2v2f2
and momentum is conserved:
m1v1i+m2v2i=m1v1f+m2v2f
If we imagine ourselves sitting on object 1 moving at velocity v1, object 2 will look like it is moving at speed v1v2. The difference in the velocities of the two objects tells us how fast object 1 is moving relative to object 2, and is sometimes called the relative velocity. If kinetic energy and momentum are conserved, we can make some predictions about the relative velocity before and after the collision.
  • The magnitude of the relative velocity is the same before and after the collision. That means if we are sitting on object 1 moving at velocity v1, object 2 will look like it is moving at the same speed both before and after the collision.
|v1iv2i|=|v1fv2f|
  • The relative velocity has opposite signs before and after the collision. If we are sitting on object 1 moving at v1, object 2 will look like it changes direction after the collision.
v1iv2i=(v1fv2f)
The relative velocity will have these properties before and after an elastic collision for any combination of masses.

Common mistakes and misconceptions

Sometimes people forget that momentum is always conserved, but only in an isolated system. If there is a net external force on the system (an external impulse), then momentum is added to the system, and momentum is not conserved.

Learn more

For an overview on momentum, read our article on the conservation of momentum.
For deeper explanations of conservation of momentum and elastic collisions, see a worked example video on bouncing fruit and an ice skater throwing a ball.
To check your understanding and work toward mastering these concepts, check out our exercise on finding speed and mass using the conservation of momentum.

Want to join the conversation?

  • piceratops tree style avatar for user eli.weber
    Wouldn't the legitimate formula be v2-v1 instead? In this formula, if the frame of reference is moving faster, the object in front appears to have a positive velocity? It seems as though this may be written incorrectly.
    (7 votes)
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    • cacteye green style avatar for user pa_u_los
      I don't understand very well what you mean. Actually that last formula is derived from the equation of conservation of kinetic energy. KA has two videos, one showing the very long way and shorter way. If you use that formula, your equations will be much less tedious and doing a system of equations with the momentum one will give you the result you are searching for this type of problems where momentum and energy is conserved (ellastic collision). The frame of reference doesn't move... I just think it's poorly worded. If you can please be more clear, if I'm capable, I would be happy to help. What drove you into that conclusion? You seem to have upvotes and so maybe I'm the only one who doesn't understand the question...
      (3 votes)
  • piceratops ultimate style avatar for user Joshua
    Let us say I am on object A moving 5 m/s in the positive direction(defined as to the right on the page in this case) and object B is moving 3 m/s also to the right and I decided to calculate relative velocity the same way in the article I would take v1( 5 m/s) and subtract v2(3 m/2) from it this leaves me with a positive 2 m/s. This means I would see object b moving to the right at a speed of 2 m/s relative to me. If you go based on intuition this can't be right as you should see object b moving away from you due to its lower speed. and it would be moving away at a velocity of -2 m/2
    (7 votes)
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    • leaf grey style avatar for user Anshari Hasanbasri
      Remember that the 2 m/s you got represents how fast you are moving (on object A) relative to object B. In other words, if your friend was on object B, she would see you moving 2 m/s away from her (not her from you).

      If you wanted to calculate how fast object B is moving away from you, then you would swap the numbers: (3-5) = -2 m/s (which satisfies your intuition).

      Hope this helps.
      (1 vote)
  • blobby green style avatar for user Shu-Mang Feng
    Im confused about relative velocity, I can see how the principle would work with objects of equal mass, but not different mass.
    (5 votes)
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    • cacteye green style avatar for user pa_u_los
      Well first of all velocity doesn't depend on the mass which is insane. Could you develop more your question? If you plug in the equations is going to give you the desired result. Remeber that we are talking about ellastic collisions where both kinetic energy and momentum are conserved.
      (2 votes)
  • blobby green style avatar for user vivian Xia
    What is the difference between impulse and momentum?
    (2 votes)
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  • blobby green style avatar for user mohamad
    Why The magnitude of the relative velocity is the same before and after the elastic collision?
    (3 votes)
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  • blobby green style avatar for user CamilleJ
    after an elastic collision, what happens to the velocity of each car?
    (2 votes)
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