Question

A small box is connected to one end of a massless spring of unstretched length $4.9m$. The other end of the spring is fixed. The system lies on a horizontal frictionless surface. The block is stretched by $0.2m$ and released from rest at $t=0$. It then executes simple harmonic motion with angular frequency $\omega =\frac{\pi }{3}rad/s$. Simultaneously at $t=0$, a small pebble is projected with speed $v$ from point $P$ at an angle of ${45}^{\circ }$ as shown in the figure. Point $P$ is a a horizontal distance of $10m$ from $O$. If the pebble hits the block at $t=1sec$, the value of $v$ is (take $g=10m/s^2$) :

• A. $\sqrt{50}m/s$
• B. $\sqrt{51}m/s$
• C. $\sqrt{52}m/s$
• D. $\sqrt{53}m/s$

Answer 1

The correct option is A $\sqrt{50}m/s$

Pebble hits the block after $1$ sec of projection

$⟹$ Pebble should reach the surface in $1$ sec

$⟹$ Time of flight of pebble $=1$ sec

Now, we know that

Time of flight $=\frac{2v\sin \theta }{g}$ ($\theta -$ angle of projection)

Thus, $\frac{2v\sin {45}^o}{g}=1$

$⟹\frac{2v}{5\omega }\times \frac{1}{\sqrt{2}}=1⟹v=5\sqrt{2}m/s$

$⟹v=\sqrt{50}m/s$