A scene in a movie has a stuntman falling through a floor onto a bed in the room

Question

A scene in a movie has a stuntman falling through a floor onto a bed in the room below. The plan is to have the actor fall on his back, but you have been hired to investigate the safety of this stunt. When you examine the mattress, you see that it effectively has a spring constant of 65144 N/m for the area likely to be impacted by the stuntman, but cannot depress more than 12.89 cm without injuring him. To approach this problem, consider a simplified version of the situation. A mass falls through a height of 3.32 m before landing on a spring of force constant 65144 N/m. Calculate the maximum mass that can fall on the mattress without exceeding the maximum compression distance.
=______________________ kg

Explanation / Answer

The max compression distance is 12.89 cm or 0.1289 m Let's calculate how much energy is stored in the mattress at this compression: U = (1/2)k(x^2) = (1/2)(65144 N/m)(0.1289 m)^2 = 541.191 J If we conserve energy, we see that this has to equal the initial gravitational potential energy: U initial = mgh h = height of object + height of mattress compression (since we will take this to be our zero point) = 3.32 m + 0.1289 m = 3.4489 m 541.191 J = m(9.8 m/s^2)(3.4489 m) m = 16.01 kg 16.01 kg is a pretty low mass so this stunt is NOT SAFE

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