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 a researcher has been hired to investigate the safety of this stunt. When the researcher examines the mattress, she sees that it effectively has a spring constant of 65144 N/m for the area likely to be impacted by the stuntman, but it cannot depress more than 12.23 cm without injuring him. To approach this problem, consider a simplified version of the situation. A mass falls through a height of 3.52 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.

maximum mass:

Answer 1

Consider that the velocity with which mass 'm' hit the mattress as v.
Initial velocity, u = 0
Distance travelled, h = 3.52 m
Acceleration, g = 9.81 m/s²
Using the formula, v² - u² = 2gh,
v² = u² + 2gh
= 0 + 2 * 9.81 * 3.52
= 69.06

The energy with which mass m hits the mattress = 1/2 mv²
Spring energy used to stop the mass, m = 1/2kx²
Where k is the spring constant and x is the compression.
1/2 mv² = 1/2 kx²
m = kx²/v²

Substituting values,
m = [65144 * 0.1223²] / 69.06
= 14.11 kg