Question

a high - jumper, having just cleared the bar, lands on an air mattress and comes to rest. had she landed directly on the hard ground, her stopping time would have been much shorter. using the impulse - momentum theorem as your guide, determine which one of the following statements is correct.
the air mattress exerts a greater impulse, and a greater net average force, on the high - jumper than does the hard ground.
the air mattress exerts a smaller impulse, and a smaller net average force, on the high - jumper than does the hard ground.
the air mattress exerts the same impulse, but a greater net average force, on the high - jumper than does the hard ground.
the air mattress exerts the same impulse, but a smaller net average force, on the high - jumper than does the hard ground.
the air mattress exerts a greater impulse, but a smaller net average force, on the high - jumper than does the hard ground.

Explanation:

Step1: Recall the impulse - momentum theorem

The impulse - momentum theorem is \(J=\Delta p\), where \(J = F_{avg}\Delta t\) (impulse \(J\) is the product of the average force \(F_{avg}\) and the time interval \(\Delta t\)) and \(\Delta p=m(v - u)\) (change in momentum, with \(m\) as mass, \(u\) as initial velocity and \(v\) as final velocity).
In both cases (landing on air - mattress and hard ground), the initial velocity \(u\) (just before landing) and final velocity \(v = 0\) (comes to rest) are the same. So, the change in momentum \(\Delta p=m(0 - u)=-mu\) is the same. Since \(J=\Delta p\), the impulse \(J\) is the same.

Step2: Analyze the force - time relationship

We know that \(J = F_{avg}\Delta t\). Rearranging for \(F_{avg}=\frac{J}{\Delta t}\).
The stopping time \(\Delta t\) on the air - mattress is larger (\(\Delta t_{air - mattress}>\Delta t_{hard - ground}\)). Since \(J\) is constant (from Step 1), and \(F_{avg}=\frac{J}{\Delta t}\), when \(\Delta t\) increases, \(F_{avg}\) decreases. So, \(F_{avg,air - mattress}

Answer:

The air mattress exerts the same impulse, but a smaller net average force, on the high - jumper than does the hard ground.