A 47.5-g golf ball is driven from the tee with an initial speed of 41.8 m/s and rises to a height of 34.5 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 8.88 m below its highest point?
a. KE = 0.5m*V^2 = 0.5m*0 = 0 J.
The velocity is zero at the max. ht.
b. V^2 = Vo^2 + 2g*h
V^2 = 41.8^2 - 19.6*(34.5-8.88)
V^2 = 1245.09
V = 35.29 m/s.
To solve this problem, we need to apply the principles of conservation of energy. The total mechanical energy of the golf ball, which includes both its kinetic energy (KE) and potential energy (PE), remains constant throughout its motion.
(a) To find the kinetic energy of the ball at its highest point, we can first find its potential energy at that point and subtract it from the initial total mechanical energy.
1. Find the potential energy at the highest point (PE):
PE = m * g * h
where m is the mass of the golf ball (47.5 g or 0.0475 kg), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height (34.5 m).
PE = 0.0475 kg * 9.8 m/s^2 * 34.5 m
2. Calculate the initial mechanical energy (Ei):
Ei = KE + PE
Since the ball starts from rest on the ground, the initial potential energy is zero. Therefore, the initial mechanical energy is equal to the initial kinetic energy:
KEi = Ei = 0.5 * m * v^2
where v is the initial speed of the ball (41.8 m/s).
Ei = 0.5 * 0.0475 kg * (41.8 m/s)^2
3. Calculate the final mechanical energy (Ef) at the highest point:
Ef = KEf + PE
Since the ball reaches its highest point, the final potential energy is equal to the initial potential energy:
PEf = PE = m * g * h
4. Finally, calculate the kinetic energy at the highest point (KEf):
KEf = Ef - PEf
= Ei - PEf
Now you can substitute the values into the formula and solve for KEf.
(b) To find the speed when the ball is 8.88 m below its highest point, we can use the same principle of conservation of energy and calculate the kinetic energy using the potential energy at that point.
1. Calculate the potential energy at 8.88 m below the highest point (PE):
PE = m * g * h
where m is the mass of the golf ball (47.5 g or 0.0475 kg), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height (8.88 m).
PE = 0.0475 kg * 9.8 m/s^2 * 8.88 m
2. Calculate the final mechanical energy (Ef) at 8.88 m below the highest point:
Ef = KEf + PE
Since the ball is not at its highest point, both kinetic energy and potential energy are present.
3. Solve for the final kinetic energy (KEf):
KEf = Ef - PE
Now, you have all the information needed to calculate the kinetic energy and speed of the ball 8.88 m below its highest point.
Remember to convert the units appropriately and apply the correct formulas in order to obtain accurate results.