Question

A small block is connected to one end of a massless spring of unstretched 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 ω = \(\frac{\pi}{3}\). Simultaneously at t = 0, a small

pebble is projected with speed v from point P at an angle of 45° as shown in the figure Point P is at a horizontal distance of 10 m from O, If the pebble hits the block at t = 1 s, the value of v is

\((a)\ \sqrt{50}\ m/s\)

\((b)\ \sqrt{51}\ m/s\)

\((c)\ \sqrt{52}\ m/s\) 

\((d)\ \sqrt{53}\ m/s\)

Answer 1

Correct option is : \((a)\ \sqrt{50}\ m/s\)

The block starts its oscillation from its positive extremum.

\(\therefore\ \)\(X_{block}(t ) =4.9 + 0.2 \cos \omega t\)

\(\therefore\ X_{block}(t = 1s) =4.9 + 0.2\cos(\frac{\pi}{3}\times1) = 5\ m\) 

\(\therefore\ \) Range of pebble =5m

So, \(\frac{v^2\sin2\theta}{g} =5\)

\(\Rightarrow\ v = \sqrt{50}\ m/s\)