A-LevelMathematicsMechanics9709/41 · May/June 2020 · Q710 Marks

A particle P of mass 0.3 kg, lying on a smooth plane inclined at 30° to the horizontal, is released from rest. P slides down the plane for a distance of 2.5 m and then reaches a horizontal plane. There is no change in speed when P reaches the horizontal plane. A particle Q of mass 0.2 kg lies at rest on the horizontal plane 1.5 m from the end of the inclined plane (see diagram). P collides directly with Q. [The diagram]

(a)It is given that the horizontal plane is smooth and that, after the collision, P continues moving in the same direction, with speed 2 m s⁻¹. Find the speed of Q after the collision.[5]
(b)It is given instead that the horizontal plane is rough and that when P and Q collide, they coalesce and move with speed 1.2 m s⁻¹. Find the coefficient of friction between P and the horizontal plane.[5]

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Mark scheme

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[M1] 0.3gsin 30 = 0.3a (a = 5) (M1 for applying Newton's second law parallel to the plane)
[M1] v² = 0 + 2 × 2.5 × a
[A1] v = 5
[M1] 0.3 × 5 + 0 = 0.3 × 2 + 0.2 w
[A1] Velocity of Q = 4.5 ms⁻¹
[M1] 0.3xz + 0 = 0.5 × 1.2
[A1] Velocity of P before collision z = 2
[B1] Friction force on P after reaches horizontal plane F = µ × 0.3 g
[M1] µ × 0.3g × 1.5 = ½ × 0.3 × 5² – ½ × 0.3 × 2²
[A1] Coefficient µ = 0.7

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