A force platform is a tool used to analyze the performance of athletes measuring the vertical force that the athlete exerts on the ground as a function of time. Starting from rest, a 60.0 kg athlete jumps down onto the platform from a height of 0.600 m. While she is in contact with the platform during the time interval 0 < t < 0.8 s, the force she exerts on it is described by the function below.
F = (9 200 N/s)t - (11 500 N/s2)t2
Assume the positive y-axis points upward.
(a) What was the athlete's velocity when she reached the platform?
(b) What impulse did the athlete receive from the platform?
(c) What impulse did the athlete receive from gravity while in contact with the platform?
(d) With what velocity did she leave the platform?
(e) To what height did she jump upon leaving the platform?
To find the answers to these questions, we need to understand the concepts of velocity, impulse, and the equations governing the motion of the athlete. Let's break it down step by step:
(a) To find the athlete's velocity when she reaches the platform, we need to calculate the area under the force-time curve. The area under the curve represents the impulse.
To find velocity, we integrate the force function over the time interval 0 to t (where t is the time when she reaches the platform):
v = ∫(0 to t) [(9,200 N/s)t - (11,500 N/s^2)t^2] dt
Evaluating this integral will give us the velocity when she reaches the platform.
(b) Impulse is defined as the change in momentum. In this case, we can calculate the impulse by multiplying the average force during the time interval by the time:
Impulse = Δp = F * ∆t
We can use the provided force function to calculate the average force and multiply it by the time interval (0 < t < 0.8s) to find the impulse received from the platform.
(c) The impulse received from gravity can be calculated using the equation:
Impulse = ∆p = m * ∆v
The athlete's mass is given as 60.0 kg. We need to find the change in velocity (∆v) while she is in contact with the platform, which we can calculate by subtracting the velocity when she first contacts the platform from the velocity when she leaves the platform.
(d) To find the velocity with which she leaves the platform, we need to find the velocity at the end of the time interval. We can use the force function to integrate and get the change in velocity (∆v) over the time interval (0 to t) and add it to the velocity at the start of the interval.
(e) Finally, to determine the height she jumps upon leaving the platform, we need to find the time it takes for her to reach the maximum height after leaving the platform. This can be done using kinematic equations. Then we can find the height using the equation for vertical displacement.
Let's calculate each of these steps one by one.