A tennis player places a 54 kg ball machine on a frictionless surface. The machine fires a 0.057 kg tennis ball horizontally with a velocity of 38 m/s toward the north. What is the final velocity of the machine?
m/s to the south
momentum is conserved.
Massmachine*veloctitymachine=-massball*velocityball
velocitymachine=-.057/54 * 38 North
and of course, -North is South
To find the final velocity of the machine, we need to apply the principle of conservation of momentum. The initial momentum of the system (which includes both the tennis ball and the machine) is equal to the final momentum of the system.
The momentum of an object can be calculated as the product of its mass and velocity (P = m * v).
Let's denote the final velocity of the machine as V_machine and the initial velocity of the tennis ball as V_tennis_ball.
Initially, the momentum of the system is given by:
Initial momentum = (mass of the tennis ball) * (initial velocity of the tennis ball) + (mass of the machine) * (initial velocity of the machine)
= (0.057 kg) * (38 m/s) + (54 kg) * (0 m/s) [since the machine is initially at rest]
= 2.166 kg·m/s + 0 kg·m/s
= 2.166 kg·m/s
According to the principle of conservation of momentum, the final momentum of the system is equal to the initial momentum.
Thus, the final momentum of the system can be expressed as:
Final momentum = (mass of the tennis ball) * (final velocity of the tennis ball) + (mass of the machine) * (final velocity of the machine)
Since the machine is placed on a frictionless surface, the horizontal velocity component of the machine remains constant throughout.
Therefore, the final velocity of the machine can be calculated as:
Final momentum = (0.057 kg) * (38 m/s) + (54 kg) * (V_machine)
2.166 kg·m/s = 2.166 kg·m/s + 54 kg·V_machine
Simplifying the equation, we get:
0 kg·m/s = 54 kg·V_machine
Dividing both sides by 54 kg, we find:
V_machine = 0 m/s
Hence, the final velocity of the machine is 0 m/s. This means that the machine remains at rest after firing the tennis ball horizontally towards the north.
To find the final velocity of the machine, we can use the principle of conservation of momentum. The momentum of the system (ball machine + tennis ball) before and after the event should remain the same.
The initial momentum of the system can be calculated as the product of the mass and the velocity of the ball:
Initial momentum = mass of the ball × velocity of the ball = 0.057 kg × 38 m/s = 2.166 kg·m/s
Since there are no external forces acting on the system, the total momentum after the event should also be 2.166 kg·m/s. However, the tennis ball is fired horizontally, meaning only the momentum in the x-direction changes.
Let's assume the final velocity of the ball machine in the x-direction is v_final-x, and in the y-direction is v_final-y. Since there is no friction and the surface is frictionless, there are no forces acting in the y-direction. This means that the velocity in the y-direction remains the same before and after the event.
Hence, v_final-y = 0 m/s.
Now, let's consider the momentum in the x-direction after the event. The mass of the ball machine is 54 kg.
Final momentum in x-direction = mass of the ball machine × final velocity in x-direction
2.166 kg·m/s = 54 kg × v_final-x
Now, we can solve for v_final-x:
v_final-x = 2.166 kg·m/s / 54 kg = 0.040 m/s
Therefore, the final velocity of the machine is 0.040 m/s to the south.
Note: In this calculation, we assumed that the collision between the ball machine and the tennis ball is elastic, meaning there is no energy loss during the collision.