A tennis player places a 60 kg ball machine on a frictionless surface, as shown below. The machine fires a 0.057 kg tennis ball horizontally with a velocity of 35.4 m/s toward the north. What is the final velocity of the machine?
To find the final velocity of the machine, we can apply the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum of an isolated system remains constant if no external forces act on it. In this case, the ball machine and the tennis ball form an isolated system since no external forces are mentioned.
The principle of conservation of momentum can be mathematically expressed as:
Total initial momentum = Total final momentum
The initial momentum of the system is zero since the ball machine is at rest. The final momentum of the system is the combined momentum of the ball machine and the tennis ball after the ball is fired.
The initial and final momentums can be calculated as follows:
Initial momentum = 0
Final momentum = (Mass of the ball machine + Mass of the tennis ball) * Final velocity
Let's calculate the final velocity:
Mass of the ball machine = 60 kg
Mass of the tennis ball = 0.057 kg
Final velocity of the ball machine = ?
Using the conservation of momentum equation:
0 = (60 kg + 0.057 kg) * Final velocity
Simplifying the equation:
0 = 60.057 kg * Final velocity
Since anything multiplied by zero is zero, we can conclude that the final velocity of the ball machine is zero.
Therefore, the final velocity of the machine is 0 m/s.
To calculate the final velocity of the machine, we can use the law of conservation of momentum. According to this law, the total momentum before the ball is fired should be equal to the total momentum after the ball is fired.
Momentum (p) is defined as the product of an object's mass (m) and its velocity (v). Mathematically, it can be expressed as:
p = m * v
Before the ball is fired, the momentum of the ball machine is zero since it's stationary. The total momentum is only due to the momentum of the ball.
After the ball is fired, the momentum of the ball machine will change. Let's assume the final velocity of the machine is v_f.
According to the law of conservation of momentum, we can write:
Initial momentum = Final momentum
(0 kg) * (0 m/s) + (0.057 kg) * (0 m/s) = (60 kg) * (v_f)
0 + 0 = 60 * v_f
0 = 60 * v_f
Dividing both sides by 60:
v_f = 0 m/s
Therefore, the final velocity of the machine is 0 m/s.