A 67.7 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 1.04 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.
Well, isn't that a dive-lemma! Let's compute the magnitude of the average force the water exerts on the diver.
First, we need to calculate the gravitational potential energy the person loses during the fall. The change in potential energy is given by:
ΔPE = mgh
Where:
m = mass of the person = 67.7 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height from which the person jumps = 3.11 m
ΔPE = (67.7 kg)(9.8 m/s^2)(3.11 m)
ΔPE ≈ 2054.35 J
Now, we can determine the work done by the water to bring the person to rest underwater. The work done is equal to the negative change in potential energy:
Work = -ΔPE
Work = -2054.35 J
Finally, we can find the magnitude of the average force using the work-energy theorem:
Work = Force × Distance
Force = Work / Distance
Given that the distance the person travels underwater is 1.04 m:
Force = -2054.35 J / 1.04 m
Force ≈ -1976.25 N
So, the magnitude of the average force that the water exerts on the diver is approximately 1976.25 Newtons. And boy, that's quite a force to make a splash!
To determine the magnitude of the average force that the water exerts on the diver, we need to calculate the total work done by this force.
1. First, let's calculate the potential energy the person has at the top of the tower:
Potential energy (PE) = mass (m) * gravitational acceleration (g) * height (h)
PE = 67.7 kg * 9.8 m/s^2 * 3.11 m
2. Next, let's calculate the potential energy the person has when submerged. This is given by the depth she reaches under the water:
Potential energy (PE submerged) = mass (m) * gravitational acceleration (g) * depth (d)
PE submerged = 67.7 kg * 9.8 m/s^2 * 1.04 m
3. The work done by the water force is equal to the change in potential energy, which is given by:
Work done = PE - PE submerged
4. Now, let's calculate the magnitude of the average force exerted by the water:
Force = Work done / distance
However, the distance in this case is the depth the person reaches, which is 1.04 m.
By substituting the values we calculated, we can find the magnitude of the average force:
Work done = (PE) - (PE submerged)
= (67.7 kg * 9.8 m/s^2 * 3.11 m) - (67.7 kg * 9.8 m/s^2 * 1.04 m)
Force = Work done / distance
= ((67.7 kg * 9.8 m/s^2 * 3.11 m) - (67.7 kg * 9.8 m/s^2 * 1.04 m)) / 1.04 m
Simplifying this further will give us the magnitude of the average force exerted by the water on the diver.
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 gravitational potential energy of the diver at the top of the tower and at the point where she comes to rest under the water's surface.
The initial potential energy (U1) can be calculated using the equation:
U1 = m * g * h1
where m is the mass of the diver (67.7 kg), g is the acceleration due to gravity (9.8 m/s^2), and h1 is the height of the tower (3.11 m).
U1 = 67.7 kg * 9.8 m/s^2 * 3.11 m
U1 = 2046.5694 Joules
The final potential energy (U2) can be calculated using the equation:
U2 = m * g * h2
where h2 is the depth of the water (1.04 m).
U2 = 67.7 kg * 9.8 m/s^2 * 1.04 m
U2 = 657.6176 Joules
The change in potential energy (ΔU) is given by:
ΔU = U2 - U1
ΔU = 657.6176 Joules - 2046.5694 Joules
ΔU = -1388.9518 Joules
Now, we can calculate the work done by the water on the diver using the equation:
Work = -ΔU
Neglecting air resistance, the work done by the water is equal to the change in potential energy.
Work = - (-1388.9518 Joules)
Work = 1388.9518 Joules
The magnitude of the average force (F) exerted by the water on the diver can be calculated using the equation:
Work = F * d
where d is the distance over which the force is applied, which is the depth of the water (1.04 m).
F = Work / d
F = 1388.9518 Joules / 1.04 m
F = 1334.1356 N
Therefore, the magnitude of the average force that the water exerts on the diver is approximately 1334.14 Newtons.