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

F_{\rm net}=mg-ky=0\quad\Rightarrow\quad y_* =\frac{mg}{k}

problems.proof.analysis

Differentiating $v^2(y)$ gives $2g-2ky/m$ , so the speed peaks where this derivative changes sign. Equivalently, the downward net force vanishes at $y_*$ .