A 48.9-g golf ball is driven from the tee with an initial speed of 45.4 m/s and rises to a height of 29.9 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 5.62 m below its highest point?
To solve this problem, we first need to understand the different types of energy involved. The total mechanical energy of an object is the sum of its kinetic energy (KE) and potential energy (PE). Kinetic energy refers to the energy of motion, while potential energy refers to the energy associated with an object's position or height.
In this problem, we are given the mass of the golf ball (48.9 g = 0.0489 kg), its initial speed (45.4 m/s), and the height it reaches (29.9 m). We are asked to find the kinetic energy at the highest point and the speed when it is 5.62 m below its highest point.
(a) To find the kinetic energy (KE) at the highest point, we know that at the highest point, the ball has reached its maximum height and its velocity is zero. Therefore, all of its initial kinetic energy has been converted into potential energy.
The potential energy (PE) at the highest point can be calculated using the formula:
PE = m * g * h
where m is the mass of the ball, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
PE = 0.0489 kg * 9.8 m/s^2 * 29.9 m
PE ≈ 14.13 J
Since the ball has no kinetic energy at its highest point, the total mechanical energy is equal to the potential energy:
Total mechanical energy = PE = 14.13 J
(b) To find the speed of the ball when it is 5.62 m below its highest point, we can use the principle of conservation of energy. The total mechanical energy is conserved (ignoring air resistance), so the sum of the kinetic energy and potential energy at any point remains constant.
At a height of 5.62 m below the highest point, we can calculate the potential energy using the same formula as above:
PE = m * g * h
PE = 0.0489 kg * 9.8 m/s^2 * 5.62 m
PE ≈ 2.67 J
Since the total mechanical energy is conserved, the sum of the kinetic energy and potential energy at this point is equal to the total mechanical energy at the highest point:
Total mechanical energy = PE + KE
KE = Total mechanical energy - PE
KE = 14.13 J - 2.67 J
KE ≈ 11.46 J
The kinetic energy of the ball when it is 5.62 m below its highest point is approximately 11.46 Joules.
To find the speed at this point, we can use the formula for kinetic energy:
KE = (1/2) * m * v^2
Rearranging the formula, we can solve for the speed (v):
v = sqrt((2 * KE) / m)
v = sqrt((2 * 11.46 J) / 0.0489 kg)
v ≈ 13.73 m/s
Therefore, the speed of the ball when it is 5.62 m below its highest point is approximately 13.73 m/s.