Question

A block of mass $m$ is projected towards a spring with velocity $v_0$. The force constant of the spring is $k$. The block is projected from a distance $\ell$ from the free end of the spring. The collision between block and the wall is completely elastic. Match the following columns :

Column I		Column II
A	Maximum compression of the spring	1	$-\sqrt{\frac{k{v}_0^2}{m}}$
B	Energy of oscillations of block	2	$\sqrt{\frac{m{v}_0^2}{k}}$
C	Time period of oscillations	3	$\frac{1}{2}m{v}_0^2$
D	Maximum acceleration of the block	4	$\left[\frac{2\ell }{v_0}+\pi \sqrt{\frac{m}{k}}\right]$

• A. (A) - (2), (B) - (3), (C) - (1), (D) - (4)
• B. (A) - (2), (B) - (3), (C) - (4), (D) - (1)
• C. (A) - (1), (B) - (4), (C) - (3), (D) - (2)
• D. (A) - (1), (B) - (2), (C) - (3), (D) - (4)

Answer 1

Answer: (B)

$\frac{1}{2}k{x}_0^2=\frac{1}{2}m{v}_0^2$ or $x_0=\sqrt{\frac{m{v}_0^2}{k}}$
Time period, $T^{\prime }=\left[2\frac{T}{4}+2t\right]$
$=\left[\frac{2\pi }{2}\sqrt{\frac{m}{k}}+2\frac{{\ell }_0}{v_0}\right]$
Energy of oscillation $=\frac{1}{2}m{v}_0^2$.