A 2.0-kg ball is travelling east at 8.0 m/s when it has a perfectly elastic head-on collision with a 3.0-kg ball traveling west at 10.0 m/s. What are the final velocities of the two balls?
To determine the final velocities of the two balls after the collision, we can use the law of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
Let's denote the initial velocity of the 2.0-kg ball as v1 and the initial velocity of the 3.0-kg ball as v2. Since the 2.0-kg ball is traveling east, its velocity v1 will be positive (+8.0 m/s). On the other hand, since the 3.0-kg ball is traveling west, its velocity v2 will be negative (-10.0 m/s).
Using the law of conservation of momentum, we can write:
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
Where:
m1 = mass of the 2.0-kg ball
m2 = mass of the 3.0-kg ball
v1 = initial velocity of the 2.0-kg ball
v2 = initial velocity of the 3.0-kg ball
v1' = final velocity of the 2.0-kg ball
v2' = final velocity of the 3.0-kg ball
Plugging in the values, we get:
(2.0 kg * 8.0 m/s) + (3.0 kg * -10.0 m/s) = (2.0 kg * v1') + (3.0 kg * v2')
Simplifying the equation:
16.0 kg·m/s - 30.0 kg·m/s = 2.0 kg·v1' - 3.0 kg·v2'
-14.0 kg·m/s = 2.0 kg·v1' - 3.0 kg·v2'
Since the collision is perfectly elastic, the total kinetic energy before and after the collision will be conserved. This means that the velocity of the 2.0-kg ball after the collision, v1', is equal to its velocity before the collision, v1, and similarly, the velocity of the 3.0-kg ball after the collision, v2', is equal to its velocity before the collision, v2.
Substituting these values into the equation:
-14.0 kg·m/s = 2.0 kg·v1 - 3.0 kg·v2
Since v1 = 8.0 m/s and v2 = -10.0 m/s, we can solve the equation:
-14.0 kg·m/s = 2.0 kg·(8.0 m/s) - 3.0 kg·(-10.0 m/s)
-14.0 kg·m/s = 16.0 kg·m/s + 30.0 kg·m/s
-14.0 kg·m/s = 46.0 kg·m/s
This equation is not valid, so it seems there may be an error in the question or the provided values. Please double-check the values and restate the question if necessary.