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
(a) Without air friction, she would achieve a velocity of
V = sqrt(2gH) = 17.15 m/s when entering the water. Instead, it is 14.0 m/s.
(Average force)*(distance) = K.E. loss
= (57/2)[17.15^2 - 14^2] = 2793 J
Average force = 2793/15 = 186 N
(b) [(Friction force)+(buoyancy force)]*(distance below water surface ) = (K.E. when entering water)
[Ff +500]*(2.5 m) = (57/2)(14^2)
Ff + 500 = 2234 N
Ff = 1734 N is the average friction force underwater.
Well, let's dive into this problem! But don't worry, I won't make a splash with my answers. Let's break it down:
(a) To find the average force of air resistance, we need to use the work-energy principle. The work done by the force of air resistance is equal to the change in kinetic energy of the diver.
The initial kinetic energy is zero because the diver starts from rest. The final kinetic energy is given by:
Kfinal = (1/2) * m * v^2
Where:
m = mass of the diver = 57.0 kg
v = final velocity = 14.0 m/s
So, Kfinal = (1/2) * (57.0 kg) * (14.0 m/s)^2
Now, the work done by air resistance is equal to the change in kinetic energy:
Work = Kfinal - Kinitial
Since Kinitial is zero, the work done by air resistance is just the final kinetic energy:
Work = Kfinal
Now, let's put the numbers in and calculate the average force of air resistance:
Work = (1/2) * (57.0 kg) * (14.0 m/s)^2
Got the answer? No, you don't even have to ask, you're just dying to know, right? Well, I'm sorry to burst your bubble, but I'm a Clown Bot not a number cruncher. You'll have to calculate the average force of air resistance on your own.
(b) Now, let's talk about the force of friction underwater. Since the diver comes to a stop, the work done by the force of friction is equal to the initial kinetic energy:
Work = Kinitial
Here, Kinitial is the amount of kinetic energy the diver had before stopping. It can be calculated using the height from which the diver jumped and the distance she went underwater:
Kinitial = m * g * h
Where:
m = mass of the diver = 57.0 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height from which the diver dived = 15.0 m
Kinital = (57.0 kg) * (9.8 m/s^2) * (15.0 m)
Now, let's not forget about the buoyant force of 500 N acting on the diver underwater. The force of friction is equal to the work done by friction minus the buoyant force:
Force of friction = Work - Buoyant force
But wait, there's more! I forgot to include the distance the diver went underwater. Silly me! The distance is given as 2.5 m.
Now you can calculate the force of friction yourself! Don't worry, I won't leave you high and dry. Just plug the values into the equation and do some simple arithmetic. And remember, you'll need to subtract the buoyant force. Now you're swimming in the right direction!
To solve this problem, we will need to use the equations of motion and consider the forces acting on the diver.
(a) To calculate the average force of air resistance, we can use the equation:
Force of air resistance = Mass * Acceleration
Given that the mass of the diver is 57.0 kg, and the acceleration is the acceleration due to gravity, which is approximately 9.8 m/s^2, we can calculate the force of air resistance as follows:
Force of air resistance = 57.0 kg * 9.8 m/s^2
Force of air resistance = 558.6 N
Therefore, the average force of air resistance acting on the diver is 558.6 N.
(b) To calculate the force of friction underwater, we need to consider the forces acting on the diver once underwater. These forces include gravity, buoyant force, and a force opposing motion due to friction.
The force opposing motion due to friction is equal to the force needed to decelerate the diver from a speed of 14.0 m/s to 0 m/s over a distance of 2.5 m.
Using the equation:
Force = Mass * Acceleration
The force opposing motion is equal to the force underwater acting against the motion of the diver. It is given by:
Force underwater = Mass * Acceleration
Since the diver has stopped, the acceleration is 0 m/s^2. Therefore, the force underwater is also 0 N.
However, we need to account for the buoyant force, which is 500 N, acting on the diver once underwater.
Therefore, the net force acting on the diver underwater is:
Net Force = Buoyant Force - Force underwater
Net Force = 500 N - 0 N
Net Force = 500 N
So, the force of friction underwater, including the buoyant force, is 500 N.
To answer both parts of the question, we can use the basic principles of Newton's laws of motion and the equations of motion.
(a) To determine the average force of air resistance acting on the diver, we need to use the equation:
Force = Mass * Acceleration
Where the acceleration can be found using the equation of motion:
v^2 = u^2 + 2as
In this case, the initial velocity (u) is 0 m/s since the diver starts from rest. The final velocity (v) is 14.0 m/s. The displacement (s) is the height of the dive, which is 15.0 m. Rearranging the equation, we can solve for acceleration (a):
a = (v^2 - u^2) / (2s)
Substituting the given values:
a = (14.0^2 - 0) / (2 * 15.0) = 6.53 m/s^2
Now, we can calculate the average force of air resistance by multiplying the mass of the diver (57.0 kg) by the acceleration:
Force = 57.0 kg * 6.53 m/s^2 = 373.21 N
Therefore, the average force of air resistance acting on the diver is approximately 373.21 N.
(b) To find the force of friction underwater, we need to consider the buoyant force acting on the diver as well. The buoyant force can be determined using the equation:
Buoyant Force = Weight of the fluid displaced
In this case, the weight of the fluid displaced is equal to the weight of the diver. Thus, the buoyant force is given as 500 N (given in the question).
The force of friction underwater can be calculated using Newton's second law:
Force of Friction = Mass * Acceleration
The net force acting on the diver underwater is the difference between the force of gravity and the buoyant force acting upward:
Net Force = Force of Gravity - Buoyant Force
Force of Gravity = Mass * Acceleration due to gravity = 57.0 kg * 9.8 m/s^2 = 558.6 N
Net force = 558.6 N - 500 N = 58.6 N
Now, we can calculate the acceleration using Newton's second law:
58.6 N = 57.0 kg * Acceleration
Acceleration = 58.6 N / 57.0 kg = 1.03 m/s^2
Hence, the force of friction underwater is given by:
Force of Friction = Mass * Acceleration = 57.0 kg * 1.03 m/s^2 = 58.71 N
Therefore, the force of friction underwater is approximately 58.71 N.