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

\boxed{x=\frac{mg+\sqrt{m^2g^2+2kmgh}}k,\quad v(y)=\sqrt{2g(h+y)-\frac{k}{m}y^2},\quad y_*=\frac{mg}{k},\quad v_{\max}=\sqrt{2gh+\frac{mg^2}{k}}}

problems.proof.analysis

These results give the maximum compression, speed as a function of compression, and the location and value of peak speed. The turning-point and maximum-speed conditions are consistent with the same energy law.

problems.proof.pitfall. Keep the sign convention consistent and retain only physically meaningful roots or directions.