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. `sqrt50m//s`
B. `sqrt(51)m//s`
C. `sqrt(52)m//s`
D. `sqrt(53)m//s`

Answer

A

Explanation

For block `x=Asin(omegat+phi)`
`t=0`, `x=Aimpliessinphi=1impliesphi=pi//2`
`x=Asin(omegat+phi)=Asin(omegat+pi//2)=Acosomegat`
At `t=1s`, `x=Acosomegaxx1=0.2cos((2pi)/(3))=0.1m`
Distance of block from `O=4.9+0.1=5m`
Range for pebble `=10-5=5m`
`R=(v^2)/(g)sin(2xx45^@)`
`5=(v^2)/(10)impliesv=sqrt(50)m//s`