19. (2 points) Ball 2, with a mass of 30kg, is moving to the left at 10m/s. With what velocity should Ball 1, with a mass of 10kg, move to the right and collide with Ball 2, so that Ball 1 rebounds with a velocity of 30m/s, and Ball 2 with a velocity of 10 m/s, after the collision? Assume the collision to be perfectly elastic.

To find the velocity with which Ball 1 should move to the right and collide with Ball 2, we can use the law of conservation of momentum. According to this law, the total linear momentum before the collision is equal to the total linear momentum after the collision in an isolated system.

The linear momentum of an object is given by the product of its mass and velocity. Therefore, we can calculate the initial momentum and final momentum of both balls and set them equal to each other.

Let's denote the velocity of Ball 1 moving to the right as v1, and the velocity of Ball 2 moving to the left as v2. Also, let's assume the positive direction as the direction to the right.

The initial linear momentum for Ball 1 (m1v1) is equal to the initial linear momentum for Ball 2 (m2v2):

m1 * v1 = m2 * v2

Substituting the given values:
10kg * v1 = 30kg * (-10m/s)

Now, let's find the final velocities after the collision. Since the collision is perfectly elastic, the relative velocities of the balls will be reversed but equal in magnitude. Therefore, we can conclude:

v1 = -30m/s (final velocity of Ball 1)
v2 = 10m/s (final velocity of Ball 2)

Now, we have enough information to solve for v1:

10kg * v1 = 30kg * (-10m/s)
10kg * v1 = -300kg·m/s
v1 = -30kg·m/s / 10kg
v1 = -30m/s

Therefore, Ball 1 should move to the right with a velocity of -30m/s (negative sign indicating that it's moving in the opposite direction).