## Illustration 8.4: Relative Velocity in Collisions

Please wait for the animation to completely load.

In this set of collisions there are no net external forces acting on the two carts. Restart. Enter new values for the velocity of each cart and the mass of the right-moving (orange) cart. Then click the "set values and play" button to register your values and play the animation **(position is given in meters and time is given in seconds)**. We have set limits on the values you can choose:

0.5 kg < m_{1} < 4 kg, 0 m/s < v_{1} < 4 m/s, and -4 m/s < v_{2} < 0 m/s.

The table gives an instantaneous reading of each cart's momentum as well as the total momentum in the two-cart system. In addition, when you select the check box, arrows representing the magnitude of the relative velocities before and after the collision between the two carts are also shown.

Because the net force on the system of two carts is zero, the change in momentum of the two-particle system is zero. In other words, momentum is conserved. Using equations, we would say that since Σ **F**_{net} **= **Δ**p/**Δt or Σ **F**_{net} **= **d**p**/dt, if the net force on a system is zero, then Δ**p/**Δt = 0 or d**p**/dt = 0, which means that the change in momentum over time must be zero. Hence the sum of the two impulses experienced by the carts must be zero. If one particle's momentum goes up, the other particle's momentum must go down by exactly the same amount.

In elastic collisions, the concept of the **relative velocity** is an important one in analyzing the collision. The relative velocity is defined as **v _{1} **-

**v**(it could also be defined as

_{2}**v**-

_{2}**v**as the choice of 1 and 2 is arbitrary).

_{1}Turn on the relative velocity arrows and vary the velocity of each cart and the mass of the right-moving (orange) cart. Determine the relationship between the relative velocity before the collision and the relative velocity after the collision. What did you find? It turns out that the magnitude of the relative velocity before and after an elastic collision is the same. However, the sign of the relative velocity changes from before to after the collision: **(v _{1} **-

**v**-

_{2})_{i}= - (v_{1}**v**

_{2})_{f}. This relationship can be verified by using the conservation of energy and conservation of momentum equations and a bit of algebra.

Consider an elastic collision where v_{1} = 1 m/s and v_{2} = -4 m/s. Clearly the relative velocity before the collision is 5 m/s. What must it be after the collision? -5 m/s. Try it and find out if this is true. Does it matter if you change the mass of the orange cart?