A tennis player places a 60 kg ball machine
on a frictionless surface. The machine fires a
0.053 kg tennis ball horizontally with a velocity of 43 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 ball machine, we need to use the principle of conservation of momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, assuming no external forces act on the system.
Let's denote the initial velocity of the ball machine as V_machine, and the final velocity of the ball machine as V'_machine. The momentum of an object is given by the product of its mass and its velocity.
The initial momentum of the ball machine is calculated as:
Momentum_initial_machine = mass_machine * V_machine
Substituting the given values:
Mass_machine = 60 kg
V_machine = 0 m/s (since the machine is at rest before firing)
Momentum_initial_machine = 60 kg * 0 m/s = 0 kg*m/s
The initial momentum of the tennis ball is given by:
Momentum_initial_ball = mass_ball * V_ball
Substituting the given values:
Mass_ball = 0.053 kg
V_ball = 43 m/s
Momentum_initial_ball = 0.053 kg * 43 m/s ≈ 2.279 kg*m/s
According to the conservation of momentum principle, the total momentum before and after the event should be equal. Therefore, we can write the equation:
Momentum_initial_machine + Momentum_initial_ball = Momentum_final_machine + Momentum_final_ball
Since the ball machine had no initial velocity (V_machine = 0 m/s), we can simplify the equation to:
Momentum_initial_ball = Momentum_final_machine + Momentum_final_ball
Plugging in the values we know:
2.279 kg*m/s = V'_machine * (mass_machine + mass_ball)
Solving for V'_machine (final velocity of the machine):
V'_machine = 2.279 kg*m/s / (mass_machine + mass_ball)
Plugging in the values:
V'_machine = 2.279 kg*m/s / (60 kg + 0.053 kg)
V'_machine ≈ 0.0379 m/s
Therefore, the final velocity of the machine is approximately 0.0379 m/s.
To find the final velocity of the machine, we need to consider the conservation of momentum.
The momentum before the tennis ball is fired is equal to the momentum after the tennis ball is fired. The momentum of an object is given by the product of its mass and velocity.
The initial momentum of the ball machine is 0 because it is at rest.
Momentum before = Momentum after
(60 kg) * (0 m/s) = (60 kg + 0.053 kg) * Vf
0 = (60.053 kg) * Vf
Therefore, the final velocity of the machine when the tennis ball is fired is 0 m/s.
There will be a recoil velocity V such that the total momentum remains zero. Ball and machine momenta are equal and opposite.
V*60 = 43*.053
V = 0.038 m/s = 3.8 cm/s