A 57.0 kg diver dives from a height of 15.0 m. She reaches a speed of 14.0 m/s just before entering the water. (a) What was the average force of air resistance (e.g., friction) acting on the diver. (b) What is the force of friction underwater if she reaches a depth of 2.5 m before stopping? Do not neglect the buoyant force of 500 N acting on the diver once underwater
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To answer these questions, we need to use the laws of motion and apply the concepts of force, weight, and acceleration. Let's break down the problem step by step:
(a) Average Force of Air Resistance:
To find the average force of air resistance acting on the diver, we can use the concept of work-energy theorem. According to this theorem, the work done by the force of air resistance is equal to the change in kinetic energy of the diver. The work done can be calculated using the formula:
Work = (1/2) * mass * (final velocity^2 - initial velocity^2)
Given:
Mass (m) = 57.0 kg
Initial velocity (vi) = 0 m/s (since the diver starts from rest)
Final velocity (vf) = 14.0 m/s
Substituting these values into the formula, we get:
Work = (1/2) * 57.0 kg * (14.0 m/s)^2
Work = 5733 J (rounded to the nearest whole number)
The work done by the air resistance is equal to the average force of air resistance multiplied by the distance traveled (which is the height of the dive, h = 15.0 m). So:
Work = Average Force of Air Resistance * Distance
5733 J = Average Force of Air Resistance * 15.0 m
Therefore, the average force of air resistance acting on the diver is:
Average Force of Air Resistance = 5733 J / 15.0 m
Average Force of Air Resistance = 382.2 N (rounded to the nearest tenth)
(b) Force of Friction Underwater:
To find the force of friction underwater, we need to consider the total force acting on the diver, which includes the weight of the diver, the buoyant force, and the force of friction underwater. We'll use Newton's second law of motion:
Total Force = Mass * Acceleration
The total force acting on the diver is the sum of the weight (mg), the buoyant force (500 N), and the force of friction underwater. We can calculate the weight using the formula:
Weight = Mass * Gravity
Weight = 57.0 kg * 9.8 m/s^2
Weight = 558.6 N (rounded to the nearest tenth)
Now, let's consider the forces acting on the diver. The upward buoyant force is equal to the weight of the displaced water, which in this case is 500 N.
Total Force = Weight + Buoyant Force + Force of Friction Underwater
Total Force = 558.6 N + 500 N + Force of Friction Underwater
Since the diver is underwater and stops at a depth of 2.5 m, this means her acceleration during this time is 0 m/s^2 (she reaches a constant speed).
Using Newton's second law, we know that Force = Mass * Acceleration. Since the acceleration is 0 m/s^2, the force of friction underwater must be equal to the total force acting on the diver.
Force of Friction Underwater = Total Force
Force of Friction Underwater = 558.6 N + 500 N
Force of Friction Underwater = 1058.6 N (rounded to the nearest tenth)
Therefore, the force of friction underwater is 1058.6 N.