Question

A box of mass m is dropped from a height H above a spring. The platform has negligible mass. The spring has spring constant k.

(a) By what amount is the spring compressed? Your answer requires the quadratic equation; to choose the correct root, note that x needs to be positive.

(b) What is the speed of the box when it is first in contact with the platform?

(c) What is the maximum speed of the box? Hint: the box speeds up until the force from the spring equals the gravitational force

Answer 1

a)

when mass of box m dropped from a height h , gravitaitonal potential energy causes the spring to compress about the distance x, so that

mgh = 0.5 k x ^2

x^2 = mgh /0.5 k

x = sqrt(2mgh/k)

b)

the speed of the box when it is first in contact with the platform is  

0.5*mv^2 = mgh

v = sqrt(2gh)

c)

the maximum speed of the box is at its equilibriun that is  

0.5*mv^2 = 0.5*k*x^2

v^2 = k*x^2/m

v = sqrt(k*x^2/m)