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) = (pi//3) rad//s`. 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 `(take g = 10 m//s^2)`.
.

Options

A. ` sqrt 50 m//s`
B. sqrt 51 m//s`
C. ` sqrt 52 m//s`
D. ` sqrt 53 m//s `

Answer

A

Explanation

For the pebble , ` u =v . Theta = 45^@, t= 1 s`
Time of flight , whn the pebble will be at horizontal level ` OX`. Where there the bloci is hit
` T= ( 2 u sin theta)/g = ( 2 v sin 45^@)/(10) = v/( 5 sqrt 2) = 1 sec`
:. ` v = 5 sqrt 2 = sqrt 50 m//s`
If the block is hit at any point (K) and (OX), then (PK= horizontal range
` = ( u^2 sin 2 theta )/g = (v^2sin 90^@)/( 10) = ( sqrt 50)^2 xx 1)/(10) = 5 m`.
This is likely, as unstretched length of sprig is ` 4.9 m` and it is stretched by ` 0.2m`.