A body of mass 1 kg collides head on elastically with a stationary body of mass 3kg. After collision, the smaller body reverses its direction of motion and moves with a speed of 2m/s. The initial speed of the smaller body before collision is __________ .
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Answer: 4
Solution
Using elastic collision, both momentum and kinetic energy are conserved.
Let initial speed of 1 kg body = u, final speed = 2 m/s (reversed), so velocity = −2 m/s.
Let final speed of 3 kg body = v.
Momentum conservation: 1·u = 1·(−2) + 3·v → u + 2 = 3v ... (1)
Energy conservation: (1/2)(1)u² = (1/2)(1)(4) + (1/2)(3)v² → u² = 4 + 3v² ... (2)
From (1): v = (u+2)/3. Substituting in (2):
u² = 4 + 3·(u+2)²/9 = 4 + (u+2)²/3
3u² = 12 + u² + 4u + 4
2u² − 4u − 16 = 0
u² − 2u − 8 = 0
(u−4)(u+2) = 0
Taking the physical solution, u = 4 m/s.
Let initial speed of 1 kg body = u, final speed = 2 m/s (reversed), so velocity = −2 m/s.
Let final speed of 3 kg body = v.
Momentum conservation: 1·u = 1·(−2) + 3·v → u + 2 = 3v ... (1)
Energy conservation: (1/2)(1)u² = (1/2)(1)(4) + (1/2)(3)v² → u² = 4 + 3v² ... (2)
From (1): v = (u+2)/3. Substituting in (2):
u² = 4 + 3·(u+2)²/9 = 4 + (u+2)²/3
3u² = 12 + u² + 4u + 4
2u² − 4u − 16 = 0
u² − 2u − 8 = 0
(u−4)(u+2) = 0
Taking the physical solution, u = 4 m/s.
[JEE Main 2023 · 25 Jan · Shift 2]