A tennis player places a 58 kg ball machine on a frictionless surface, as shown below. The machine fires a 0.068 kg tennis ball horizontally with a velocity of 31.6 m/s toward the north. What is the final velocity of the machine?
______m/s to the south
V=3m/s
To find the final velocity of the machine, we can use the principle of conservation of momentum. The total momentum before the ball is fired is equal to the total momentum after the ball is fired.
Let's denote the final velocity of the machine as v.
The initial velocity of the machine is 0 (since it is at rest).
The momentum before the ball is fired is given by:
Momentum_before = (mass of ball + mass of machine) * initial velocity of machine
The momentum after the ball is fired is given by:
Momentum_after = mass of machine * final velocity of machine
Using the principle of conservation of momentum:
Momentum_before = Momentum_after
Therefore:
(mass of ball + mass of machine) * initial velocity of machine = mass of machine * final velocity of machine
Plugging in the given values:
(0.068 kg + 58 kg) * 0 = 58 kg * v
Simplifying:
(58.068 kg) * 0 = 58 kg * v
0 = 58 kg * v
Since anything multiplied by zero is zero, we can conclude that the final velocity of the machine is 0 m/s.
To find the final velocity of the machine, we can use the principle of conservation of momentum. The total momentum before the tennis ball is fired is equal to the total momentum after the ball is fired.
Before the ball is fired, the machine is at rest, so its initial velocity is 0 m/s. The ball has a velocity of 31.6 m/s toward the north.
Let's assume the final velocity of the machine is π£π to the south. Since the machine has a mass of 58 kg and was at rest initially, the momentum of the machine before the ball is fired is 0 kgβ
m/s.
The momentum of the ball before it is fired is given by:
πππππ Γ π£ππππ = (0.068 kg) Γ (31.6 m/s) = 2.1248 kgβ
m/s toward the north.
Using the law of conservation of momentum, we can equate the total momentum before and after the ball is fired:
0 kgβ
m/s + 2.1248 kgβ
m/s (initial momentum of the machine and ball) = 58 kg Γ π£π (final momentum of the machine).
Simplifying this equation, we get:
2.1248 kgβ
m/s = 58 kg Γ π£π.
Now, we can solve for π£π by isolating it on one side of the equation:
π£π = 2.1248 kgβ
m/s / 58 kg.
π£π β 0.0367 m/s to the south.
Therefore, the final velocity of the machine is approximately 0.0367 m/s to the south.
Total momentum remains zero.
Ball momentum = - (machine momentum)
0.068*31.6 = -58 Vmachine
Solve for Vmachine.
The minus sign means that it goes south.
It is called the recoil velocity.
In the real world, friction would keep the ball machine from slipping.