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

x=\frac{mg+\sqrt{m^2g^2+2kmgh}}{k};\qquad h=0\Rightarrow x=\frac{2mg}{k}

problems.proof.analysis

The quadratic has one positive and one negative root; only the positive root is a physical compression. If release is from the spring top, $2mg/k$ is consistent with falling through the compression distance before stopping.