A pitcher throws a 0.140-kg baseball, and it approaches the bat at a speed of 50.0 m/s. The bat does Wnc = 65.0 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 25.0 m above the point of impact.
To determine the speed of the ball after it leaves the bat and is 25.0 m above the point of impact, we can use the principle of conservation of mechanical energy.
The principle of conservation of mechanical energy states that the total mechanical energy of a system is conserved if there are no external forces doing work on the system. In this case, we can assume that there are no external forces (such as air resistance) doing work on the ball.
The total mechanical energy of the ball can be divided into two parts: kinetic energy (KE) and potential energy (PE).
Initially, the ball only has kinetic energy, which is given by the equation:
KE = (1/2) * m * v^2
Where m is the mass of the ball (0.140 kg) and v is the initial velocity of the ball (50.0 m/s).
The work done by the bat on the ball (Wnc) is equal to the change in the ball's total mechanical energy. In this case, the work done by the bat is positive because it is doing work on the ball:
Wnc = ΔKE + ΔPE
ΔKE is the change in the ball's kinetic energy, which is given by:
ΔKE = KE_final - KE_initial
Since the ball initially has only kinetic energy:
ΔKE = KE_final - (1/2) * m * v^2
The change in potential energy (ΔPE) is equal to the negative of the work done by gravity (mgh), where g is the acceleration due to gravity (9.8 m/s^2):
ΔPE = -m * g * h
In this case, the ball is 25.0 m above the point of impact:
ΔPE = -0.140 kg * 9.8 m/s^2 * 25.0 m
Substituting these values into the conservation of mechanical energy equation, we get:
Wnc = ΔKE + ΔPE
65.0 J = KE_final - (1/2) * 0.140 kg * (50.0 m/s)^2 - (0.140 kg * 9.8 m/s^2 * 25.0 m)
Now, we can rearrange the equation and solve for KE_final:
KE_final = (1/2) * 0.140 kg * (50.0 m/s)^2 + 0.140 kg * 9.8 m/s^2 * 25.0 m + 65.0 J
After calculating the right-hand side of the equation, we can find the final kinetic energy (KE_final). Finally, to find the final velocity (v_final) of the ball, we can use the equation for kinetic energy:
KE_final = (1/2) * m * v_final^2
Solving this equation for v_final will give us the speed of the ball after it leaves the bat and is 25.0 m above the point of impact.
If you assume all this is elastic..
KEf=KEi+work done-changePE
KEi from initial speed; work done given, change PE=mg(25)
figure KE final. The, calculate Vf