Question

Two blocks $A\left(5kg\right)$ and $B\left(2kg\right)$ attached to the ends of a 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. when the extension of the spring is maximum, the velocities of $A$ and $B$ are zero
• B. the maximum extension of the spring is 25 cm
• C. when the extension of the spring is maximum, the velocities of $A$ and $B$ are maximum
• D. the maximum compression occur for the first time after $\frac{\pi }{56}sec$

Answer 1

The correct option is B the maximum extension of the spring is 25 cm

The maximum elongation occurs when the velocity of both the particles becomes equal.
Since there is no external force acting in the horizontal direction (for the blocks + spring system), thus the linear momentum is conserved in horizontal direction:
If the final common velocity is $V$
$5\times 3+2\times 10=(5+2)VV=5m{s}^{-1}$
Also applying law of conservation of energy for the system (work done by non-conservative force is zero):
Initial Energy $=$ Final Energy
$\frac{1}{2}\times 5\times 3^2+\frac{1}{2}\times 2\times {10}^2=\frac{1}{2}\times 1120x^2+\frac{1}{2}\times 7\times 5^2$
Solving the above equation we get $x=25$ cm
Also it can be observed that the compression and expansion occur alternately.