Question

Two blocks of masses $1kg$ and $3kg$ are moving with velocities $2m/s$ and $1m/s$ respectively, as shown. If the spring constant is $75N/m$, the maximum compression (in cm) of the spring is

Answer 1

Due to spring force, velocity of $1kg$ decreases and that of $3kg$ increases. At the maximum compression, both the blocks move with same velocity.


By the momentum conservation
$1\times 2+3\times 1=(1+3)v$

$v=\frac{5}{4}m/s$

By the energy conservation

$\frac{1}{2}\times 1(2{)}^2+\frac{1}{2}\times 3(1{)}^2=\frac{1}{2}(1+3)v^2+\frac{1}{2}k{x}^2$

$\frac{7}{2}=2{\left(\frac{5}{4}\right)}^2+\frac{75}{2}{x}^2$

$\frac{75}{2}{x}^2=\frac{7}{2}-\frac{25}{8}=\frac{3}{8}$

$x^2=\frac{1}{100}$

$x=\frac{1}{10}m=10cm$