A 66 kg diver steps off a 13 m tower and drops from rest straight down into the water. If he comes to rest 5.0 m beneath the surface, determine the average resistance force exerted on him by the water.
To determine the average resistance force exerted on the diver by the water, we can use the concept of work-energy theorem.
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done on the diver by the water is equal to the change in the diver's kinetic energy as he moves from the top of the tower to his final position 5.0 m beneath the surface.
The initial kinetic energy of the diver is zero since he is at rest, and the final kinetic energy is also zero since he comes to rest. Therefore, the work done on the diver is equal to the change in potential energy as he falls.
The potential energy of an object near the surface of the Earth is given by the formula: PE = mgh, where m is the mass of the object (66 kg), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height (13 m).
So, the initial potential energy of the diver when he is at the top of the tower is PE_initial = mgh = (66 kg)(9.8 m/s^2)(13 m).
The final potential energy of the diver when he is 5.0 m beneath the surface is PE_final = mgh = (66 kg)(9.8 m/s^2)(5.0 m).
The work done on the diver is then given by the difference in potential energy: W = PE_final - PE_initial.
Finally, we can calculate the average resistance force exerted by the water using the work-energy theorem: W = Fd, where F is the average resistance force and d is the displacement. Since the displacement is 18 m (13 m + 5 m), we can solve for F.
F = W / d.
Substituting the values we have calculated, we can find the average resistance force exerted on the diver by the water.
To determine the average resistance force exerted on the diver by the water, we can use the work-energy principle.
The work-energy principle states that the net work done on an object is equal to its change in kinetic energy. In this case, the net work done on the diver is the sum of the work done by gravity and the work done by the water resistance.
1. First, let's calculate the change in potential energy:
Change in potential energy = mgh
= (66 kg)(9.8 m/s²)(13 m)
= 8,226 J
2. Next, let's calculate the change in kinetic energy:
Change in kinetic energy = 1/2 mv² - 1/2 mu²
Since the diver starts from rest, the initial velocity (u) is 0.
Therefore, the change in kinetic energy is given by:
Change in kinetic energy = 1/2 mv²
= 1/2 (66 kg)(v)²
3. Using the work-energy principle, we can equate the change in potential energy and the change in kinetic energy to find the velocity (v) of the diver just before reaching the water's surface:
8,226 J = 1/2 (66 kg)(v)²
4. Solve the equation for v:
(66 kg)(v)² = 2(8,226 J)
(66 kg)(v)² = 16,452 J
(v)² = 16,452 J / (66 kg)
(v)² = 249 J/kg
v = √(249 J/kg)
v ≈ 15.8 m/s
5. To calculate the average resistance force exerted on the diver by the water, we can use the equation for drag force:
Drag force = (1/2)ρAv²C
where ρ is the density of water, A is the cross-sectional area of the diver perpendicular to the motion, and C is the drag coefficient.
6. Since the diver is in free fall, the drag force must balance the weight of the diver:
Drag force = weight of the diver
(1/2)ρAv²C = mg
7. Rearrange the equation for the drag force to solve for the average resistance force:
Average resistance force = (1/2)ρAv²C
= (1/2)(1000 kg/m³)(A)(15.8 m/s)²(C)
To find the cross-sectional area (A) and drag coefficient (C), more information about the shape and orientation of the diver is needed.