A 86.0 kg diver falls from rest into a swimming pool from a height of 6.30 m. It takes 1.86 s for the diver to stop after entering the water. Find the magnitude of the average force exerted on the diver during that time.
To find the magnitude of the average force exerted on the diver, we need to use Newton's second law of motion, which states that the force (F) acting on an object is equal to its mass (m) multiplied by its acceleration (a).
First, let's find the acceleration of the diver during the time it takes to stop. We can use the kinematic equation:
v = u + at
Where:
v is the final velocity (which is 0 m/s because the diver has stopped),
u is the initial velocity (which is 0 m/s because the diver falls from rest),
a is the acceleration of the diver,
and t is the time taken to stop (1.86 s).
Plugging in the values, we can solve for a:
0 = 0 + a * 1.86
Simplifying the equation gives us:
a = 0 / 1.86
a = 0 m/s²
Since the acceleration is 0 m/s², we know that the net force acting on the diver is also 0 N at the moment of stopping.
However, we are asked for the magnitude of the average force exerted on the diver during the time it takes to stop. We know that a force is required to decelerate the diver, so the net force acting on the diver before it stops must be nonzero.
We can assume that the force acting on the diver is constant throughout the deceleration process. Therefore, we can calculate the average force by dividing the change in momentum by the time interval.
The change in momentum can be calculated using the equation:
Change in momentum = m * (final velocity - initial velocity)
The final velocity is 0 m/s (since the diver has stopped), and the initial velocity can be calculated using the equation of motion:
s = ut + (1/2)at^2
Where:
s is the height (6.30 m),
u is the initial velocity (unknown),
t is the time taken to stop (1.86 s), and
a is the acceleration of the diver (0 m/s²).
Rearranging the equation, we get:
u = (s - (1/2)at^2) / t
Plugging in the values, we can calculate the initial velocity:
u = (6.30 - (1/2)(0)(1.86)^2) / 1.86
u = (6.30 - 0) / 1.86
u = 6.30 / 1.86
u = 3.387 m/s
Now we can calculate the change in momentum:
Change in momentum = m * (0 - u)
Change in momentum = 86.0 kg * (0 - 3.387 m/s)
Change in momentum = -291.482 kg⋅m/s
Since the average force can be calculated as the change in momentum divided by the time interval:
Average force = Change in momentum / t
Average force = -291.482 kg⋅m/s / 1.86 s
Average force = -156.656 N
Therefore, the magnitude of the average force exerted on the diver during the time it takes to stop is approximately 156.656 N.