A pitcher throws a 0.140-kg baseball, and it approaches the bat at a speed of 35.0 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 initial mechanical energy of the system is given by the sum of the kinetic energy and the potential energy of the ball at the point of impact. The final mechanical energy is given by the sum of the kinetic energy and the potential energy of the ball when it is 25.0 m above the point of impact.

The initial mechanical energy (Ei) is the sum of the kinetic energy (Ek) and the potential energy (Ep) at the point of impact:
Ei = Ek + Ep

The final mechanical energy (Ef) is the sum of the kinetic energy (Ek') and the potential energy (Ep') when the ball is 25.0 m above the point of impact:
Ef = Ek' + Ep'

Since the work done by the bat on the ball (Wnc) is equal to the change in mechanical energy (ΔE) of the system, we have:
Wnc = Ef - Ei

Substituting the expressions for Ef and Ei, we get:
Wnc = (Ek' + Ep') - (Ek + Ep)

Now, let's calculate the initial kinetic energy (Ek) and potential energy (Ep) at the point of impact.

Given:
Mass of the baseball (m) = 0.140 kg
Initial speed of the ball (vi) = 35.0 m/s

The initial kinetic energy (Ek) is given by the formula:
Ek = (1/2) * m * v^2
= (1/2) * 0.140 kg * (35.0 m/s)^2

The initial potential energy (Ep) is given by:
Ep = m * g * h
= 0.140 kg * 9.8 m/s^2 * 0 m

Since the ball is at the point of impact, its height (h) is zero, and thus, the potential energy (Ep) is zero.

Therefore, the initial mechanical energy (Ei) is just equal to the initial kinetic energy (Ek):
Ei = Ek

Now, let's calculate the final kinetic energy (Ek') and potential energy (Ep') when the ball is 25.0 m above the point of impact.

Given:
Final height of the ball (h') = 25.0 m

The final potential energy (Ep') is given by:
Ep' = m * g * h'
= 0.140 kg * 9.8 m/s^2 * 25.0 m

The final kinetic energy (Ek') can be derived by rearranging the equation Wnc = (Ek' + Ep') - (Ek + Ep):
Ek' = Wnc + Ek + Ep - Ep'
= 65.0 J + Ek + 0 J - Ep'

Finally, to calculate the final speed of the ball (vf), we use the formula for kinetic energy:
Ek' = (1/2) * m * v'^2

Rearranging the equation, we get:
vf = √((2 * Ek') / m)

Now, plug in the values and calculate the final speed (vf).