A pitcher throws a 0.111-kg baseball, and it approaches the bat at a speed of 37.4 m/s. The bat does 78.6 J of work on the ball in hitting it. Ignoring air resistance, determine the speed of the ball after the ball leaves the bat and is 30.8 m above the point of impact.
To determine the speed of the ball after it leaves the bat and is 30.8 m above the point of impact, we can use the principle of conservation of energy.
The initial energy of the ball-bat system is the sum of the kinetic energy (K) and the potential energy (U):
Initial Energy = K + U
Since the ball is approaching the bat at a speed of 37.4 m/s, its initial kinetic energy (K1) can be calculated using the formula:
K1 = (1/2) * mass * velocity^2
Using the given mass of the baseball (0.111 kg) and the initial speed (37.4 m/s), we can substitute these values into the formula:
K1 = (1/2) * 0.111 kg * (37.4 m/s)^2
Next, we need to calculate the potential energy (U1) at the point of impact. The potential energy at a height h can be calculated using the formula:
U1 = mass * gravitational acceleration * height
Substituting the values of mass (0.111 kg), gravitational acceleration (9.8 m/s^2), and height (0 m), we get:
U1 = 0.111 kg * 9.8 m/s^2 * 0 m
Since the ball is at a height of 30.8 m above the point of impact after leaving the bat, we can calculate its potential energy (U2):
U2 = mass * gravitational acceleration * height
Substituting the values of mass (0.111 kg), gravitational acceleration (9.8 m/s^2), and height (30.8 m), we get:
U2 = 0.111 kg * 9.8 m/s^2 * 30.8 m
According to the principle of conservation of energy, the final energy of the ball-bat system is equal to the initial energy. The final energy is again the sum of the kinetic energy (K2) and the potential energy (U2):
Final Energy = K2 + U2
From the given information, we know that the bat does 78.6 J of work on the ball. This work is equal to the change in kinetic energy, so we can equate it to the difference between the final and initial kinetic energies:
Work = K2 - K1
Solving for K2:
K2 = Work + K1
Substituting the given value of work (78.6 J) and the calculated value of K1, we can find K2.
Finally, to find the speed of the ball after it leaves the bat and is 30.8 m above the point of impact, we can use the formula for kinetic energy:
K2 = (1/2) * mass * velocity^2
Solving for velocity:
velocity = sqrt((2 * K2) / mass)
By substituting the calculated value of K2 and the given mass (0.111 kg) into the formula, we can find the speed of the ball after it leaves the bat.