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Physical Quantities

Three carts m...

Question

Three carts move on a frictionless track with inertias and velocities as shown. The carts collide and stick together after successive collisions.
1) Find loss of mechanical energy when B & C stick together.
2)Find magnitude of impulse experienced by A when it sticks to combined mass (B&C)

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Solution

1) By conservation of linear momentum,
$m_{2} v_{2} + m_{3} v_{3} = (m_{2} + m_{3}) v_{3} \Rightarrow 1 \times 1 + 2 (- 2) = 3 \times v_{3} ∴ v_{4} = - 1 ㎨$
$L o s s o f M . E = {K E}_{I n i t i a l} - {K E}_{F i n a l} = \frac{1}{2} m_{2} v_{2}^{2} + \frac{1}{2} m_{3} v_{3}^{2} - \frac{1}{2} {(m}_{2} + m_{3}) v_{4}^{2} = \frac{1}{2} \times 1 \times 1^{2} + \frac{1}{2} \times 2 \times {(- 2)}^{2} - \frac{1}{2} \times 3 \times {(- 1)}^{2} = 3 J$

2) $m_{1} v_{1} + (m_{2} + m_{3}) v_{4} = (m_{1} + m_{2} + m_{3}) v_{5} ∴ v_{5} = \frac{2 + 3 (- 1)}{5} = - \frac{1}{5} ㎧$

$∴$ Impulse=change in momentum
$= m_{1} v_{1} - (m_{2} + m_{3}) v_{5} = 2 \times 1 + 5 \times (- \frac{1}{5}) = 1 N s$

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