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. What was the average force of air resistance (e.g., friction) acting on the diver?
Bonus: 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.
To find the average force of air resistance acting on the diver, we can use the principles of energy conservation. The initial potential energy of the diver is converted into kinetic energy just before entering the water.
1. Calculate the initial potential energy (PE) of the diver using the formula:
PE = mass × gravity × height
PE = 57.0 kg × 9.8 m/s² × 15.0 m
2. Calculate the final kinetic energy (KE) of the diver just before entering the water using the formula:
KE = 0.5 × mass × velocity²
KE = 0.5 × 57.0 kg × (14.0 m/s)²
3. The difference between the initial potential energy and the final kinetic energy represents the work done by air resistance, which is the average force of air resistance (since force is work done per distance).
Average Force of Air Resistance = PE - KE
For the bonus question, we need to calculate the force of friction underwater using the buoyant force and the weight of the diver.
4. Calculate the gravitational force acting on the diver:
Weight = mass × gravity
Weight = 57.0 kg × 9.8 m/s²
5. Subtract the buoyant force from the gravitational force to find the net force acting on the diver:
Net Force = Weight - Buoyant Force
Now we have the net force acting on the diver, but we need to convert it into the force of friction.
6. The force of friction is equal in magnitude and opposite in direction to the net force acting on the diver:
Force of Friction (underwater) = - Net Force
By following these steps, we can determine both the average force of air resistance acting on the diver and the force of friction underwater.
To find the average force of air resistance acting on the diver, we can use the laws of motion.
Step 1: Calculate the diver's gravitational potential energy before diving using the formula:
Potential Energy = mass * gravity * height
Potential Energy = 57.0 kg * 9.8 m/s^2 * 15.0 m
Potential Energy = 8331 J
Step 2: Calculate the diver's kinetic energy just before entering the water using the formula:
Kinetic Energy = 0.5 * mass * velocity^2
Kinetic Energy = 0.5 * 57.0 kg * (14.0 m/s)^2
Kinetic Energy = 5484 J
Step 3: The work done against air resistance is equal to the difference between the initial potential energy and final kinetic energy:
Work against air resistance = Potential Energy - Kinetic Energy
Work against air resistance = 8331 J - 5484 J
Work against air resistance = 2847 J
Step 4: We know that work done is equal to the force multiplied by the distance moved in the direction of the force. So, the average force of air resistance can be calculated using the formula:
Work = Force * distance
Force = Work / distance
Force = 2847 J / 15.0 m
Force = 189.8 N
Therefore, the average force of air resistance acting on the diver is approximately 189.8 N.
Bonus: To find the force of friction underwater, we need to consider the buoyant force acting on the diver. The force of friction underwater can be found using Newton's second law.
Step 1: Calculate the net force acting on the diver underwater. The net force is the difference between the buoyant force and the force opposing motion (friction).
Net Force = Buoyant Force - Force of Friction
Given:
Buoyant Force = 500 N
Depth = 2.5 m
Step 2: Calculate the weight of the diver using the formula:
Weight = mass * gravity
Weight = 57.0 kg * 9.8 m/s^2
Weight = 558.6 N
Step 3: Calculate the force of friction using Newton's second law:
Net Force = mass * acceleration
Net Force = Weight - Force of Friction
Rearranging the equation, we have:
Force of Friction = Weight - Net Force
Given:
Depth = 2.5 m
Step 4: Calculate the net force using the work-energy principle:
Net Force = Work done / distance
Given that the diver comes to rest underwater, the work done against the net force is equal to the diver's potential energy at that depth:
Potential Energy = mass * gravity * depth
Potential Energy = 57.0 kg * 9.8 m/s^2 * 2.5 m
Potential Energy = 1381.5 J
Net Force = Potential Energy / depth
Net Force = 1381.5 J / 2.5 m
Net Force = 552.6 N
Step 5: Calculate the force of friction using the equation:
Force of Friction = Weight - Net Force
Force of Friction = 558.6 N - 552.6 N
Force of Friction = 6 N
Therefore, the force of friction underwater is approximately 6 N (neglecting other external forces).