Question

Two blocks $A\left(5kg\right)$ and $B\left(2kg\right)$ are attached to the ends of a spring of spring constant $1120N/m$ are placed on a smooth horizontal plane with the spring undeformed. Simultaneously velocities of $3m/s$ and $10m/s$ along the line of the spring in the same direction are imparted to $A$ and $B$ then,
(a) find the maximum extension of the spring
(b) when does the first maximum compression occurs after start.

Answer 1

maximum extension is seen when relative seperation stops, i.e velocity of both block will become same $\left(v\right)$.

$\left(a\right)$According to momentum conservation,

$5\times 3+2\times 10=(5+2)v$

$v=5m{s}^{-1}$

Applying energy conservation,

$△KE=$Work done by all forces

$\frac{1}{2}(5+2)5^2=\frac{1}{2}\times 5\times 9-\frac{1}{2}\times 2\times$ $100=-\dfrac1{2}kx^2$

$20=k{x}^2$

$x=\sqrt{\frac{20}{1120}}$

$x=\sqrt{\frac{1}{56}}$

$\left(b\right)$Maximum compression will occur when velocity of approach between both blocks will become zero.