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 77144 N/m for the area likely to be impacted by the stuntman, but cannot depress more than 14.65 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 77144 N/m. Calculate the maximum mass that can fall on the mattress without exceeding the maximum compression distance.

Explanation / Answer

Given ,

Spring constant = 77144 N/m

And it cannot depress more than 14.65 cm without injuring him.

So maximum allowed compression =xmax = 14.65 cm = 0.1465 m

A mass falls through a height of 3.32 m

before landing on a spring of force constant 77144 N/m

Let m be the mass

When falls from rest thruogh aheight of 3.32 m ,

g = 10m/s^2

So Total Energy = Potential energy of mass= mgh = m * g * 3.32 m

Elastic potential energy stored in spring = 1/2 * k *x^2

Total energy is converted from Potential energy of the mass to Elastic potential energy of the Spring when it compresses maximum.

So, equation we get is

mgh = m * g * 3.32 m = 1/2 * k * x^2

x must be maximum for maximum mass .

So x max = 0.1465 m

m max= [ 77144 * (0.1465)^2 ] / [2 * 10 * 3.32]

m max= 1655.68 / 66.4

m max = 24.934 Kg

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