A block of mass 2kg is sitting on a vertical spring, compressing it by 4 cm. Another block of mass 1kg is dropped on it from a height of 45 cm. The two block stick together after collision. What is the maximum compression of the spring (in cm) ?

Engineering

Physics

Collision

Question

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$2 g = {kx}_{0} = k \times \frac{4}{100}$

k = 500 N/m

$v = \sqrt{2 gh} = \sqrt{2 \times 10 \times 0 .45} = 3 m / s$

1 × 3 = (2 + 1)v
v = 1 m/s

at maximum compression

$3 g \times 2 y + \frac{1}{2} \times 500 [(0 .04)^{2} - (0 .04 + y)^{2}] = 0 - \frac{1}{2} \times 3 \times 1^{2}$

$30 y + 250 \times [\frac{10}{10^{4}} - [\frac{10}{10^{4}} + y^{2} + 0 .08 y]] = - \frac{3}{2}$

$30 y - 250 y^{2} - 20 y = - \frac{3}{2}$

$250 y^{2} - 10 y - \frac{3}{2} = 0$

$y = \frac{10 \pm \sqrt{100 + 4^{2} \times \frac{3}{2} \times 250}}{500} = 0 .1$

maximum compression = 14 cm

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