Question 6.5: Let’s Play Pool Goal Solve an elastic collision in one dimen...
Let’s Play Pool Goal Solve an elastic collision in one dimension.
Problem Two billiard balls of identical mass move toward each other as in Active Figure 6.11. Assume that the collision between them is perfectly elastic. If the initial velocities of the balls are +30.0 cm/s and -20.0 cm/s, what is the velocity of each ball after the collision? Assume friction and rotation are unimportant.
Strategy Solution of this problem is a matter of solving two equations, the conservation of momentum and conservation of energy equations, for two unknowns, the final velocities of the two balls. Instead of using Equation 6.11 for conservation of energy, use Equation 6.14, which is linear, hence easier to handle.
\frac{1}{2} m_{1} v_{1} i^{2}+\frac{1}{2} m_{2} v_{2 i}^{2}=\frac{1}{2} m_{1} v_{1 f}^{2}+\frac{1}{2} m_{2} v_{2} f^{2}
v_{1 i}-v_{2 i}=-\left(v_{1 f}-v_{2 f}\right)
Write the conservation of momentum equation. Because m_{1}=m_{2} , we can cancel the masses, then substitute v_{1 i}=+30.0 m / s \text { and } v_{2 i}=-20.0 cm / s (Step 3):
\begin{aligned}m_{1} v_{1 i}+m_{2} v_{2 i} &=m_{1} v_{1 f}+m_{2} v_{2 f} \\30.0 cm / s +(-20.0 cm / s ) &=v_{1 f}+v_{2 f} \\10.0 cm / s &=v_{1 f}+v_{2 f} (1)\end{aligned}
Next, apply conservation of energy in the form of Equation 6.14 (Step 4):
\begin{aligned}v_{1 i}-v_{2 i} &=-\left(v_{1 f}-v_{2 f}\right) \\30.0 cm / s -(-20.0 cm / s ) &=v_{2 f}-v_{1 f} \\50.0 cm / s &=v_{2 f}-v_{1 f} (2)\end{aligned}
Now solve (1) and (2) simultaneously (Step 5):
v_{1 f}=-20.0 cm / s \quad v_{2 f}=+30.0 cm / s
Remarks Notice the balls exchanged velocities—almost as if they’d passed through each other. This is always the case when two objects of equal mass undergo an elastic head-on collision.