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A small block is connected to one end of a massless spring of un-stretched length \( 4.9 \mathrm{~m} \). The other end of the spring
\( \mathrm{P} \) (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by \( 0.2 \mathrm{~m} \) and
W released from rest at \( t=0 \). It then executes simple harmonic motion with angular frequency \( \omega=\frac{\pi}{3} \mathrm{rad} / \mathrm{s} \).
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 \( 10 \mathrm{~m} \) from \( O \). If the pebble hits the block at \( t=1 \mathrm{~s} \), the value of \( v \) is (take \( g=10 \mathrm{~m} / \mathrm{s}^{2} \) )
[IIT-JEE 2012]
(a) \( \sqrt{50} \mathrm{~m} / \mathrm{s} \)
(b) \( \sqrt{51} \mathrm{~m} / \mathrm{s} \)
(c) \( \sqrt{52} \mathrm{~m} / \mathrm{s} \)
(d) \( \sqrt{53} \mathrm{~m} / \mathrm{s} \)
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