The carts in figure are sliding to the right at 1.0 m/s on a smooth level ground. The spring between them has a spring constant of 120 N/m and is compressed 40 cm. The carts slide past a flame that burns through the string holding them together. The carts are not attached to the spring. Afterward, what is the speed (in m/s) of 300g cart after it loses contact with the spring.

Engineering

Physics

Momentum and Energy Conservation for System of Particles

Question

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V_cm = 1 m/s

In the C-frame, $\frac{1}{2} {kx}^{2} = \frac{1}{2} \times 0 .1 v_{1}^{2} + \frac{1}{2} \times 0 .3 \times v_{2}^{2}$

$120 \times 0 {.4}^{2} = 0 .1 v_{1}^{2} + 0 .3 v_{2}^{2}$ ....... (i)

0.1v₁ + 0.3v₂ = 0 ....... (ii)
⇒ v₁ = –3v₂

$120 \times \frac{16}{100} = \frac{1}{10} \times 9 v_{2}^{2} + \frac{3}{10} v_{2}^{2}$

$12 \times 16 = 12 v_{2}^{2}$

v₂ = 4 m/s
∴ velocity of 300 g cart in ground frame = 5 m/s