A 45.0 kg diver steps off a 8.0 m high diving board and drops straight down into the water. If the diver comes to rest 5.0 m below the surface of the water, determine 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 need to consider the work done on the diver.
The work done on an object is defined as the product of the force exerted on the object and the distance over which the force is applied. In this case, the force exerted by the water will oppose the downward motion of the diver.
The work done by the resistance force can be calculated using the formula:
Work = Force × Distance
The gravitational potential energy of the diver just before entering the water is given by:
Potential Energy = mass × gravitational acceleration × height
Since the diver comes to rest 5.0 m below the surface of the water, the work done by the resistance force can be calculated as:
Work = Potential Energy
Substituting the given values:
Work = (45.0 kg) × (9.8 m/s²) × (8.0 m - 5.0 m)
Simplifying this equation gives:
Work = 45.0 kg × 9.8 m/s² × 3.0 m
Now, we have the work done by the resistance force. To find the average resistance force, we divide the work by the distance over which the force is applied.
Average Resistance Force = Work / Distance
Average Resistance Force = (45.0 kg × 9.8 m/s² × 3.0 m) / 3.0 m
Simplifying this equation gives:
Average Resistance Force = 45.0 kg × 9.8 m/s²
Using the value for gravitational acceleration g:
Average Resistance Force = 441 N
Therefore, the average resistance force exerted on the diver by the water is 441 N.
To determine the average resistance force exerted on the diver by the water, we can use the principle of work-energy. The work-energy principle states that the work done on an object is equal to the change in its kinetic energy.
First, let's find the initial potential energy of the diver when they step off the diving board. The potential energy (PE) is given by the equation PE = mgh, where m is the mass (45.0 kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the diving board (8.0 m).
PE = 45.0 kg × 9.8 m/s² × 8.0 m = 3528 J
Next, let's find the final potential energy of the diver when they come to rest 5.0 m below the water's surface. The potential energy is given by the equation PE = mgh, where m is the mass (45.0 kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the distance below the surface (5.0 m).
PE = 45.0 kg × 9.8 m/s² × 5.0 m = 2205 J
The change in potential energy is equal to the negative work done by the resistance force. So, we can calculate the work done by the resistance force using the equation:
Work = PE_initial - PE_final
Work = 3528 J - 2205 J = 1323 J
Finally, we can determine the average resistance force by dividing the work done by the distance traveled. Since the diver drops vertically straight down, the distance traveled is equal to the height of the diving board (8.0 m).
Average resistance force = Work / Distance
Average resistance force = 1323 J / 8.0 m
Average resistance force = 165.375 N
Therefore, the average resistance force exerted on the diver by the water is approximately 165.375 N.
well, there is a rick you can use here to make it easy. If the force is constant, then the average speed is halfway between the starting speed and the final speed (which is zero here.
So what is the speed when he hits the water?
v final = sqrt (2 g h) =sqrt(2*9.81 * 8)
= 12.5 m/s
NOW the slow down
net force up = m a = rate of change of momentum
I will assume neutral buoyancy so weight down = displacement force up
then
net force up = water drag
how long to stop?
average speed in water = 6.26 m/s
so time to stop = 5m/6.26m/s = .798 second
change in momentum = 45 * 12.5 = 562.5 kg m/s
(NOTE - this ignores the "added mass" of the water. Water around the diver is accelerated with the diver and thus requires more force in real life)
so in the end
Faverage = 562.5 / .798 = 705 Newtons
another way (easier)
The force F times 5 meters = potential energy lost in water
= m g (height above water)
note no pe change below water because buoyancy is about the same as weight
5 F = 45 * 9.81 * 8
F = 706 Newtons