A small block is connected to one end of a massless spring of un-stretched length $4.9 \ m$. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by $0.2$ $m$ 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 at a horizontal distance of $10 \ cm$ from $O$. If the pebble hits the block at $t=1 \ s$, the value of $v$ is (take $g =10 \ m / s ^2$ )
IIT 2012, Advanced
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Time of flight for projectile
$T =\frac{2 u \sin \theta}{ g }=1 sec . $
$\frac{2 u \sin 45}{g}=1 sec \text {. } $
$u=\frac{g}{\sqrt{2}} $
$u=\sqrt{50} \ m / s $
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