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, $T^{\prime}=\left[2 \frac{T}{4}+2 t\right]$
$=\left[\frac{2 \pi}{2} \sqrt{\frac{m}{k}}+2 \frac{\ell_0}{v_0}\right]$
Energy of oscillation $=\frac{1}{2} m v_0^2$.