Problem 1

A small object of mass

m

is held at rest a height

h

above the free top of a light vertical spring of stiffness

k

standing on a horizontal floor. It is released, falls vertically, and contacts the spring; neglect air resistance and all losses, take compression as positive, and let gravity be

g

. Let

x

denote the greatest compression. (a) Find

x

and check the limit

h=0

. (b) Find the object's velocity when the compression is

y

, where

0\le y\le x

. (c) Find the compression at which its speed is greatest and that maximum speed.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\frac12mv^2+\frac12ky^2=m g(h+y)\quad\Rightarrow\quad v(y)=\sqrt{2g(h+y)-\frac{k}{m}y^2}

problems.proof.analysis

At compression $y$ , the object has descended $h+y$ and the spring stores $ky^2/2$ . Conservation of energy gives the squared speed; take the positive root during the downward motion.