A lift starts at t=0 with upward acceleration aₗ=λt. At the same instant, a ball is projected vertically upward inside the lift with speed u relative to the lift. The non-zero time after which the ball returns to the thrower's hand is:
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Question

A lift starts at t=0 with upward acceleration aₗ=λt. At the same instant, a ball is projected vertically upward inside the lift with speed u relative to the lift. The non-zero time after which the ball returns to the thrower's hand is:

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G1 Physics · Accepted Answer

The lift acceleration is aₗ = λt. Relative acceleration of the ball with respect to the lift is relative acceleration = −g − λt. With relative initial velocity u, relative velocity = u − gt − λt²/2. Relative displacement: relative displacement = ut − gt²/2 − λt³/6. For return to the thrower's hand, relative displacement=0. Ignoring the trivial root t=0, u − gt/2 − λt²/6 = 0. Multiplying by 6: λt² + 3gt − 6u = 0. The positive root is t = [−3g+√(9g²+24λu)]/(2λ).