A long spring is kept vertical with its lower end attaches to the end attached to the ground and the upper end free. A block is released above the spring so that it strikes the top of spring after falling through a distance of 40 cm as shown. As the block compresses the spring, assume that the spring remains always vertical. Mass of the block is 1 kg and spring constant is 100 N/m. Choose the correct options. $({Take g=10m/s}^{2})$ .

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Maximum compression of the spring is 10 cm

Maximum compression of the spring is 40 cm

Velocity of the block is maximum just before striking the spring

Velocity of the block is maximum when the compression in spring is 10 cm

answer is B, D.

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Detailed Solution

Velocity of block is maximum when
$m g = k x \Rightarrow x = \frac{10}{100} = 0.1 m$
Compression is maximum when velocity of block is zero.
Applying energy conservation
$m g (h + x) = \frac{1}{2} k x^{2}$
$\Rightarrow 10 (0.4 + x) = 50 x^{2}$
$\Rightarrow 50 x^{2} - x - 0.4 = 0$
$x = \frac{1 \pm \sqrt{1 + 18}}{10} = \frac{1 + 3}{10} = 0.4 m$