A 47.0 g golf ball is driven from the tee with an initial speed of 60.0 m/s and rises to a height of 24.4 m.
(a) Neglect air resistance and determine the kinetic energy of the ball at its highest point.
(b) What is its speed when it is 7.0 m below its highest point?
To solve this problem, we'll need to use the principles of conservation of energy. The total mechanical energy of the golf ball is conserved as long as we neglect air resistance. The total mechanical energy is the sum of the kinetic energy and the potential energy.
(a) To find the kinetic energy of the ball at its highest point, we need to determine the potential energy at that point and subtract it from the initial total mechanical energy.
1. First, let's calculate the potential energy at the highest point. The potential energy formula for an object near the surface of the Earth is given by:
Potential Energy (PE) = mass * gravity * height
PE = (47.0 g) * (9.8 m/s^2) * (24.4 m)
2. Now, let's calculate the initial total mechanical energy. The total mechanical energy is the sum of the kinetic energy and potential energy at the initial point.
Total Mechanical Energy (E) = Kinetic Energy (KE) + Potential Energy (PE)
At the initial point, the potential energy is zero, so the total mechanical energy is equal to the kinetic energy:
E = KE_initial
KE_initial = E = (1/2) * mass * velocity^2 = (1/2) * (47.0 g) * (60.0 m/s)^2
3. Finally, subtract the potential energy at the highest point from the initial total mechanical energy to find the kinetic energy at the highest point:
KE_highest = KE_initial - PE
(b) To find the speed of the ball when it is 7.0 m below its highest point, we'll use the principle of conservation of mechanical energy again. This time, we'll subtract the potential energy at the given height from the initial total mechanical energy and solve for the kinetic energy at that point. Then, we can find the speed corresponding to that kinetic energy.
1. Calculate the potential energy at the given height (7.0 m) using the same formula as before:
PE = (47.0 g) * (9.8 m/s^2) * (7.0 m)
2. Subtract the potential energy at that height from the initial total mechanical energy to find the kinetic energy:
KE_given_height = KE_initial - PE
3. Finally, calculate the speed corresponding to the kinetic energy at the given height using the kinetic energy formula:
KE_given_height = (1/2) * mass * velocity^2
Solve for velocity:
velocity = sqrt((2 * KE_given_height) / mass)
By following these steps and plugging in the given values, you can find the answers to both parts (a) and (b) of the problem.