Example from Physics, 8.3 Elastic and Inelastic Collisions

Calculating Final Velocity: Elastic Collision of Two Carts

Two hard, steel carts collide head-on and then ricochet off each other in opposite directions on a frictionless surface (see Figure 8.10). Cart 1 has a mass of 0.350 kg and an initial velocity of 2 m/s. Cart 2 has a mass of 0.500 kg and an initial velocity of −0.500 m/s. After the collision, cart 1 recoils with a velocity of −0.94 m/s. What is the final velocity of cart 2?

An illustration shows before and after diagrams of two carts moving toward each other on a frictionless surface. The cart on the left is labeled m one and the car on the right is labeled m two. Both diagrams are labeled System of Interest. In the before diagram, a velocity vector, v one, points from the cart on the left to the cart on the right. A second, shorter velocity vector, v two, points from the cart on the right to the cart on the left. Two equations are shown: p one plus p one equals p total and net F equals zero. In the after diagram, the velocity vectors are equal in magnitude and point away from the carts. They are labeled v one prime and v two prime. The equation p one prime plus p one prime equals p total is shown.

Figure 8.10 Two carts collide with each other in an elastic collision.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
Since the track is frictionless, F_net = 0 and we can use conservation of momentum to find the final velocity of cart 2.
Solution
As before, the equation for conservation of momentum for a one-dimensional elastic collision in a two-object system is

$m_{1} v_{1} + m_{2} v_{2} = m_{1} v^{'}_{1} + m_{2} {v^{'}}_{2} .$
8.12

The only unknown in this equation is v′₂. Solving for v′₂ and substituting known values into the previous equation yields

$\begin{matrix} {v^{'}}_{2}^{} & = & \frac{m_{1} v_{1} + m_{2} v_{2} - m_{1} {v^{'}}_{1}^{}}{m_{2}} \\ = & \frac{(0 .350 kg) (2 .00 m/s) + (0 .500 kg) (- 0 .500 m/s) - (0 .350 kg) (- 0 .94 m/s)}{0 .500 kg} \\ = & 1 .56 m/s. \end{matrix}$
8.13

0 of 2 steps open

How did it go?

Strategy

Solution

Strategy

Solution