A tennis player places a 64 kg ball machine on a frictionless surface. The machine fires a 0.051 kg tennis ball horizontally with a velocity of 36 m/s toward the north. What is the final velocity of the machine? Answer in units of m/s

To find the final velocity of the machine, we need to analyze the momentum conservation in this scenario.

Momentum is defined as the product of mass and velocity. In an isolated system, the total momentum before an event is equal to the total momentum after the event.

Given:
Mass of the ball machine, m1 = 64 kg
Mass of the tennis ball, m2 = 0.051 kg
Initial velocity of the tennis ball, v2 = 36 m/s (toward the north)

The initial velocity of the ball machine, v1, is not provided.

Let's assume the initial velocity of the ball machine is v1 (magnitude) and its direction is south.

According to the law of conservation of momentum, the initial momentum should be equal to the final momentum.

Initial momentum (before the event) = Final momentum (after the event)

Final momentum of the ball machine (m1 * vf1) + Final momentum of the tennis ball (m2 * vf2) = Initial momentum of the system (m1 * v1 + m2 * v2)

Since the ball machine is placed on a frictionless surface, its final velocity (vf1) would be zero. This is because there are no external forces acting on the machine.

So, the equation becomes:
m1 * vf1 + m2 * vf2 = m1 * v1 + m2 * v2

Plugging in the given values:
64 kg * 0 m/s + 0.051 kg * vf2 = 64 kg * v1 + 0.051 kg * 36 m/s

Rearranging the equation:
0.051 kg * vf2 = 64 kg * v1 + 0.051 kg * 36 m/s

Dividing both sides by 0.051 kg:
vf2 = (64 kg * v1 + 0.051 kg * 36 m/s) / 0.051 kg

Simplifying the equation:
vf2 = 64 kg * v1 + 36 m/s

The final velocity of the machine is vf2, which is the same as the velocity of the tennis ball since the machine doesn't have any horizontal motion. Therefore, the final velocity of the machine is 36 m/s.