A 53.0-kg person jumps from rest off a 12.0-m-high tower straight down into the water.
Neglect air resistance. She comes to rest 3.00 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, well, well, looks like we have a diver with a splashy situation! Let's dive into this physics problem, shall we?
Since we're neglecting air resistance, we don't have to worry about any unwanted guests crashing the party. Now, we need to determine the magnitude of the average force that the water exerts on the diver.
To do that, we can use the concept of work. The work done on an object by a force is equal to the force applied multiplied by the distance over which the force is applied.
In this case, the distance the diver travels underwater is (12.0 m + 3.00 m) = 15.0 m. So, the work done by the water force is equal to the force applied multiplied by 15.0 m.
But we also know that the work done on the diver-loss is equal to the negative change in potential energy, as the diver comes to rest. The change in potential energy (ΔPE) is equal to the initial potential energy (mgh) minus the final potential energy (mgh), where m is the mass of the diver, g is the acceleration due to gravity, and h is the height.
Now, let's put our party hats on and do some number crunching!
ΔPE = mgh_initial - mgh_final
Where m = 53.0 kg, g = 9.8 m/s², h_initial = 12.0 m, and h_final = -3.00 m (negative because the diver is under the water surface).
ΔPE = (53.0 kg)(9.8 m/s²)(12.0 m) - (53.0 kg)(9.8 m/s²)(-3.00 m)
Now, the work-energy principle states that the work done on an object is equal to its change in energy. So, we can equate the work done by the water force to the change in potential energy.
Force_applied × 15.0 m = (53.0 kg)(9.8 m/s²)(12.0 m) - (53.0 kg)(9.8 m/s²)(-3.00 m)
Finally, let's solve for the force_applied, because nobody likes to be left hanging at a physics party!
Force_applied = [(53.0 kg)(9.8 m/s²)(12.0 m) - (53.0 kg)(9.8 m/s²)(-3.00 m)] / 15.0 m
Now, my dear friend, it's time for you to put some numbers into this equation and calculate the magnitude of that average force that the water exerts on the diver. Just remember one thing: don't ruin the party by forgetting to include the minus sign for the final potential energy because the diver is under the water surface. Happy calculations!
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 total energy of the system is conserved, so the potential energy they lose while falling is equal to the work done on them by the water.
The potential energy lost is given by the equation:
Potential Energy Lost = m * g * h
where
m = mass of the person (53.0 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height of the tower (12.0 m)
Potential Energy Lost = 53.0 kg * 9.8 m/s^2 * 12.0 m
Next, we need to calculate the work done on the person by the water. This is given by the equation:
Work Done = Force * Distance
where
Force = average force exerted by the water on the diver
Distance = distance submerged in the water (3.00 m)
Work Done = Average Force * Distance
Now, using the principle of conservation of energy, we can equate the potential energy lost to the work done:
Potential Energy Lost = Work Done
53.0 kg * 9.8 m/s^2 * 12.0 m = Average Force * 3.00 m
Now, we can solve for the average force:
Average Force = (53.0 kg * 9.8 m/s^2 * 12.0 m) / 3.00 m
Average Force = 53.0 * 9.8 * 12 / 3
Calculating this expression, we find that the magnitude of the average force that the water exerts on the diver is approximately 2079.2 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 initial potential energy of the person at the top of the tower. The potential energy is given by the formula:
PE = m * g * h
where m is the mass of the person (53.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the tower (12.0 m).
PE = 53.0 kg * 9.8 m/s^2 * 12.0 m
PE = 6,211.2 J
This is the amount of potential energy the person has before jumping.
Now, let's calculate the final potential energy of the person when they come to rest 3.00 m under the surface of the water. The final potential energy is given by the formula:
PE = m * g * h
where h is the depth the person goes into the water (3.00 m).
PE = 53.0 kg * 9.8 m/s^2 * 3.00 m
PE = 1,558.2 J
Next, we need to find the work done by the water on the diver. The work is given by the formula:
Work = PE_initial - PE_final
Work = 6,211.2 J - 1,558.2 J
Work = 4,653 J
Since work is equal to force multiplied by the distance, we can use this to find the average force exerted by the water on the diver:
Work = Force * distance
4,653 J = Force * 3.00 m
To find the force:
Force = 4,653 J / 3.00 m
Force ≈ 1551 N
Therefore, the magnitude of the average force that the water exerts on the diver is approximately 1551 N.