10.4 Elastic Collisions in One Dimension

A collision is said to be elastic if the total kinetic energy before and after the collision is the same. In other words, kinetic energy is conserved. The mass  is called the projectile and mass  is called the target. A one dimensional collision is also called a head-on collision.


Blue particle Red particle
mblue= kg mred= kg
v1i= m/s v2i= m/s
xblue= m xred= m


Here there are two situations:

Here the target is stationary (  ) and the projectile is moving with velocity .

From conservation of the total kinetic energy we have


From conservation of total linear momentum and we have

If we know the masses of the two objects and the initial velocity, we can solve the two equations for final velocities. We get:

We can see that the target always moves into the same direction into which the projectile started. However,  can be positive or negative depending on the values of  and  ).

We have three special cases here.

·             (equal mass)

  and .

The projectile stops, while the target moves on with the original velocity of the projectile.

·              (massive target)

  and .

The projectile (small) ball simply bounces off the target (large ball), while the large ball moves at very small speed.

·             (massive projectile)


Here both the target and the projectile are moving. If the masses and the initial velocities of the projectile and target are known, then we can calculate the final velocities of the target and the projectile using

Since there are NO external forces during a collision, the velocity of the center of mass before and after the collision is the same and is given by:

Demos/Animations Examples/Checkpoints Interactive Problems
am = animation ,dm = demo, cp = checkpoint, ex = example, ip = interactive problem