A 65 kg diver stands still on a tower, 10.0 m above the water.
(a) Calculate the gravitational potential energy of the diver, relative to the height of the water. Assume the gravitational field constant is 9.8 N/kg
9.8
N
/
kg
(b) The diver reaches the surface of the water at a speed of 14 m/s
14
m
/
s
. Calculate the diver’s kinetic energy.
(c) Compare your answers to (a) and (b). Explain your comparison using concepts learned in this course. Assume there is no air resistance during the dive.
(d) Calculate the speed of the diver 5.0 m
5.0
m
above the water.
(a) The gravitational potential energy of the diver can be calculated using the formula:
Potential energy = mass × gravitational field constant × height
Substituting the given values:
Potential energy = 65 kg × 9.8 N/kg × 10.0 m
= 6370 J
So, the gravitational potential energy of the diver, relative to the height of the water, is 6370 J.
(b) The kinetic energy of the diver can be calculated using the formula:
Kinetic energy = 0.5 × mass × velocity^2
Substituting the given values:
Kinetic energy = 0.5 × 65 kg × (14 m/s)^2
= 6370 J
So, the diver's kinetic energy is also 6370 J.
(c) The comparison between the gravitational potential energy and the kinetic energy of the diver reveals that they are equal. This is consistent with the conservation of energy principle, which states that energy cannot be created or destroyed, only transferred or transformed. In this case, the potential energy of the diver at height is transformed into kinetic energy as the diver falls. The fact that both values are equal confirms that energy has been conserved in the system.
(d) To calculate the speed of the diver 5.0 m above the water, we can use the conservation of mechanical energy.
Initial mechanical energy (at 10.0 m) = Final mechanical energy (at 5.0 m)
Initial mechanical energy = Potential energy at 10.0 m
= 65 kg × 9.8 N/kg × 10.0 m
= 6370 J
Final mechanical energy = Potential energy at 5.0 m + Kinetic energy at 5.0 m
Potential energy at 5.0 m = 65 kg × 9.8 N/kg × 5.0 m
= 3185 J
Using the equation:
Initial mechanical energy = Final mechanical energy
6370 J = 3185 J + Kinetic energy at 5.0 m
Solving for the kinetic energy at 5.0 m:
Kinetic energy at 5.0 m = 6370 J - 3185 J
= 3185 J
To calculate the speed using the kinetic energy formula:
Kinetic energy = 0.5 × mass × velocity^2
3185 J = 0.5 × 65 kg × velocity^2
Solving for velocity:
velocity^2 = (2 * 3185 J) / (65 kg)
= 97.769 m^2/s^2
Taking the square root:
velocity = √(97.769 m^2/s^2)
= 9.888 m/s
So, the speed of the diver 5.0 m above the water is approximately 9.888 m/s.