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
(b) The diver reaches the surface of the water at a speed of 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
above the water.
(4 points)
(a) To calculate the gravitational potential energy of the diver, we use the formula:
Gravitational potential energy = mass * gravity constant * height
Plugging in the values:
Gravitational potential energy = 65 kg * 9.8 N/kg * 10.0 m
Gravitational potential energy = 6370 J
The gravitational potential energy of the diver is 6370 J.
(b) To calculate the kinetic energy of the diver, we use the formula:
Kinetic energy = 0.5 * mass * velocity^2
Plugging in the values:
Kinetic energy = 0.5 * 65 kg * (14 m/s)^2
Kinetic energy = 6370 J
The diver's kinetic energy is also 6370 J.
(c) The answers to parts (a) and (b) are the same. This is explained by the principle of conservation of mechanical energy. In this case, as there is no air resistance, the only energy involved is gravitational potential energy and kinetic energy. As the diver falls, gravitational potential energy is converted into kinetic energy. At the height of the water, all the potential energy is converted into kinetic energy, resulting in the same value for both.
(d) To calculate the speed of the diver 5.0 m above the water, we can use the conservation of mechanical energy:
Gravitational potential energy + Kinetic energy = constant
At the height of the water, the kinetic energy is 6370 J. Therefore:
Gravitational potential energy + 6370 J = constant
Gravitational potential energy = mass * gravity constant * height
Gravitational potential energy = 65 kg * 9.8 N/kg * 5.0 m
Gravitational potential energy = 3185 J
Therefore, the constant is 3185 J + 6370 J = 9555 J.
Using this constant and the formula for kinetic energy, we can solve for the velocity:
Kinetic energy = 0.5 * mass * velocity^2
9555 J = 0.5 * 65 kg * velocity^2
velocity^2 = 9555 J / (0.5 * 65 kg)
velocity^2 = 293.08 m^2/s^2
Taking the square root of both sides:
velocity = √293.08 m^2/s^2
velocity = 17.1 m/s
Therefore, the speed of the diver 5.0 m above the water is 17.1 m/s.
(a) To calculate the gravitational potential energy of the diver relative to the height of the water, we can use the formula:
Gravitational Potential Energy = mass x gravity x height
Given:
mass = 65 kg
gravity = 9.8 N/kg
height = 10.0 m
Plugging in the values, we have:
Gravitational Potential Energy = 65 kg x 9.8 N/kg x 10.0 m
Gravitational Potential Energy = 6,370 J
Therefore, the gravitational potential energy of the diver relative to the height of the water is 6,370 Joules.
(b) To calculate the diver's kinetic energy, we can use the formula:
Kinetic Energy = 0.5 x mass x speed^2
Given:
mass = 65 kg
speed = 14 m/s
Plugging in the values, we have:
Kinetic Energy = 0.5 x 65 kg x (14 m/s)^2
Kinetic Energy = 6,205 J
Therefore, the diver's kinetic energy is 6,205 Joules.
(c) Comparing the gravitational potential energy (6,370 J) and the kinetic energy (6,205 J) of the diver, we can see that the potential energy is slightly greater than the kinetic energy. This can be explained by the conservation of energy principle. As the diver falls, potential energy is converted into kinetic energy. The slight difference in values can be attributed to factors such as rounding errors or the conversion of potential energy into other forms of energy, like air resistance.
(d) To calculate the speed of the diver 5.0 m above the water, we can use the conservation of energy principle.
Initial potential energy = final potential energy + final kinetic energy
Given:
mass = 65 kg
gravity = 9.8 N/kg
initial height = 10.0 m
final height = 5.0 m
Let's assume the final kinetic energy is K.
At the initial height:
Gravitational Potential Energy = 65 kg x 9.8 N/kg x 10.0 m
Initial potential energy = 6,370 J
At the final height:
Gravitational Potential Energy = 65 kg x 9.8 N/kg x 5.0 m
Final potential energy = 3,185 J
Using the conservation of energy principle:
Initial potential energy = final potential energy + final kinetic energy
6,370 J = 3,185 J + K
Rearranging the equation:
K = 6,370 J - 3,185 J
K = 3,185 J
Therefore, the speed of the diver 5.0 m above the water is equivalent to the kinetic energy at that height, which is 3,185 Joules.
To answer these questions, we need to apply the concepts of gravitational potential energy and kinetic energy.
(a) To calculate the gravitational potential energy, we can use the formula:
Gravitational Potential Energy (GPE) = mass * gravitational field constant * height
Given:
mass (m) = 65 kg
height (h) = 10.0 m
gravitational field constant (g) = 9.8 N/kg
Substituting the values into the formula, we get:
GPE = 65 kg * 9.8 N/kg * 10.0 m
GPE = 6370 J
Therefore, the gravitational potential energy of the diver relative to the height of the water is 6370 J.
(b) To calculate the kinetic energy, we can use the formula:
Kinetic Energy (KE) = 0.5 * mass * velocity^2
Given:
mass (m) = 65 kg
velocity (v) = 14 m/s
Substituting the values into the formula, we get:
KE = 0.5 * 65 kg * (14 m/s)^2
KE = 6370 J
Therefore, the diver's kinetic energy is also 6370 J.
(c) Comparing the answers to (a) and (b), we can see that the gravitational potential energy and kinetic energy of the diver are equal. This is consistent with the law of conservation of energy, which states that energy cannot be created or destroyed, only transferred or transformed. In this scenario, as the diver falls from the tower, the potential energy is converted into kinetic energy, and since there is no air resistance, the energy remains constant. This conservation of energy principle is an important concept in physics.
(d) To calculate the speed of the diver 5.0 m above the water, we can use the concept of conservation of energy. At this point, the diver still has some gravitational potential energy, which will be converted into kinetic energy as they fall the remaining 5.0 m.
Using the formula for kinetic energy:
Kinetic Energy (KE) = 0.5 * mass * velocity^2
We know the mass (65 kg) and the remaining gravitational potential energy of the diver (GPE) can be calculated using the formula from part (a) by subtracting the GPE at 10.0 m from the GPE at 5.0 m.
GPE at 5.0 m = mass * gravitational field constant * height
GPE at 5.0 m = 65 kg * 9.8 N/kg * 5.0 m
GPE at 5.0 m = 3185 J
Since GPE is equal to the remaining KE:
KE = 3185 J
Now we can solve for the velocity:
KE = 0.5 * mass * velocity^2
3185 J = 0.5 * 65 kg * velocity^2
velocity^2 = (3185 J) / (0.5 * 65 kg)
velocity^2 = 98 m^2/s^2
Taking the square root of both sides, we get:
velocity = √(98 m^2/s^2)
velocity ≈ 9.90 m/s
Therefore, the speed of the diver when they are 5.0 m above the water is approximately 9.90 m/s.