A 49.5-g golf ball is driven from the tee with an initial speed of 41.9 m/s and rises to a height of 29.7 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 5.66 m below its highest point?
To determine the kinetic energy of the golf ball at its highest point (a), we can use the principle of conservation of energy. The total mechanical energy of the ball remains constant throughout its motion, neglecting air resistance. The mechanical energy consists of potential energy (PE) due to height and kinetic energy (KE) due to motion.
(a) We'll start by calculating the potential energy (PE) of the ball at its highest point. The potential energy is given by the formula:
PE = m * g * h
Where:
m is the mass of the ball (49.5 g = 0.0495 kg),
g is the acceleration due to gravity (approximately 9.8 m/s²),
h is the height of the ball (29.7 m).
So, substituting the given values into the formula, we have:
PE = 0.0495 kg * 9.8 m/s² * 29.7 m
PE ≈ 14.35 J
At the highest point, the potential energy is at its maximum, which means the kinetic energy is zero. Therefore, at the highest point, the kinetic energy (KE) of the ball is also zero.
(b) To determine the speed of the ball when it is 5.66 m below its highest point, we can use the principle of conservation of energy again. This time, we'll consider the change in potential energy (PE) and kinetic energy (KE) as the ball moves from the highest point to a lower position.
The kinetic energy at any point is given by the formula:
KE = (1/2) * m * v²
Where:
m is the mass of the ball,
v is the velocity of the ball.
Since the ball starts from rest at its highest point, the initial kinetic energy is zero. Using the conservation of energy, we can equate the initial potential energy to the final potential energy and final kinetic energy:
PE_initial = KE_final
m * g * h_initial = (1/2) * m * v²
We can cancel out the mass (m) from both sides of the equation. Rearranging the equation, we get:
v = √(2 * g * h_initial)
Substituting the given values into the formula, we have:
v = √(2 * 9.8 m/s² * 5.66 m)
v ≈ 11.26 m/s
Therefore, the speed of the golf ball when it is 5.66 m below its highest point is approximately 11.26 m/s.