A 65.2 kg person jumps from rest off a 3.11 m-high tower straight down into the water. Neglect air resistance during the descent. She comes to rest 1.08 m under the surface of the water. Determine the magnitude of the average force that the water exerts on the diver. This force is nonconservative.
To determine the magnitude of the average force that the water exerts on the diver, we can use the principle of conservation of energy.
The initial potential energy of the person on the tower is given by the equation:
PE_initial = m * g * h
where m is the mass of the person (65.2 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the tower (3.11 m).
PE_initial = (65.2 kg) * (9.8 m/s^2) * (3.11 m)
PE_initial = 2018.4516 Joules
The final potential energy of the person underwater can be calculated using the equation:
PE_final = m * g * d
where d is the depth the person has submerged (1.08 m).
PE_final = (65.2 kg) * (9.8 m/s^2) * (1.08 m)
PE_final = 690.0096 Joules
The work done by the water on the person can be determined as the difference in potential energy:
Work done = PE_final - PE_initial
Work done = 690.0096 Joules - 2018.4516 Joules
Work done = -1328.442 Joules
The negative sign indicates that work is done on the person by the water.
Now, we can calculate the force exerted by the water using the equation:
Force = Work done / distance
Since we are looking for the average force, the distance is equal to the depth the person submerged:
Force = -1328.442 Joules / 1.08 m
Force = -1230.237 N
The magnitude of the average force that the water exerts on the diver is approximately 1230.237 N.
To determine the magnitude of the average force that the water exerts on the diver, we can use the principle of conservation of energy.
First, let's calculate the potential energy of the person at the top of the tower:
Potential energy = mass * gravity * height
= 65.2 kg * 9.8 m/s^2 * 3.11 m
= 2,023.98 Joules
Next, let's calculate the potential energy of the person when they reached a depth of 1.08 m under the surface of the water:
Potential energy = mass * gravity * height
= 65.2 kg * 9.8 m/s^2 * 1.08 m
= 708.2784 Joules
The difference in potential energy between the top of the tower and the water surface is equal to the work done by the water on the person. So, we have:
Work = Change in potential energy
= Potential energy at the top - Potential energy at the water surface
= 2,023.98 J - 708.2784 J
= 1,315.7016 Joules
Now, since the force is nonconservative, it dissipates as work is done on the body by the water. The work done on the person is equal to the negative of the work done by the water on the person. So, the magnitude of the average force that the water exerts on the diver is equal to:
Magnitude of force = -work done on the person / distance traveled
= -1,315.7016 Joules / 1.08 m
= -1,218.52 N
Therefore, the magnitude of the average force that the water exerts on the diver is 1,218.52 N, directed upwards. Note that the negative sign indicates that the force is in the opposite direction to the displacement.