A pitcher throws a 0.140-kg baseball, and it approaches the bat at a speed of 45.0 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.
-34.3 J and 45.93m/s
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 ball includes its kinetic energy as it approaches the bat and its potential energy due to its height above the point of impact. The final mechanical energy will consist of the kinetic energy of the ball after it leaves the bat and its potential energy at a height of 25.0 m.
The equation for the conservation of mechanical energy can be written as follows:
Initial kinetic energy + Initial potential energy = Final kinetic energy + Final potential energy
The initial kinetic energy (K1) can be calculated using the equation:
K1 = 0.5 * mass * velocity^2
where mass is the mass of the baseball (0.140 kg) and velocity is its initial speed (45.0 m/s).
K1 = 0.5 * 0.140 kg * (45.0 m/s)^2
The initial potential energy (U1) is given by:
U1 = mass * gravity * height
where mass is the mass of the baseball (0.140 kg), gravity is the acceleration due to gravity (9.8 m/s^2), and height is the initial height above the point of impact (0 m).
U1 = 0.140 kg * 9.8 m/s^2 * 0 m
The final potential energy (U2) is given by:
U2 = mass * gravity * height
where mass is the mass of the baseball (0.140 kg), gravity is the acceleration due to gravity (9.8 m/s^2), and height is the final height above the point of impact (25.0 m).
U2 = 0.140 kg * 9.8 m/s^2 * 25.0 m
Since the work done on the ball by the bat is equal to the change in mechanical energy, we can substitute the work (Wnc = 65.0 J) for the difference in mechanical energy:
Wnc = K2 + U2 - K1 - U1
where K2 is the final kinetic energy of the ball after it leaves the bat.
Rearranging the equation, we can solve for K2:
K2 = Wnc + K1 + U1 - U2
Now we have determined the final kinetic energy, which we can convert to the final speed of the ball using the equation for kinetic energy:
K2 = 0.5 * mass * velocity^2
Solving for velocity, we get:
velocity = sqrt(2 * K2 / mass)
Substituting K2 and the mass of the baseball (0.140 kg) into the equation, we can calculate the final speed of the ball after it leaves the bat and is 25.0 m above the point of impact.