A-LevelMathematicsDifferential equations9709/52 · Feb/Mar 2017 · Q68 Marks
O and A are fixed points on a rough horizontal surface, with OA = 1 m. A particle P of mass 0.4 kg is projected horizontally with speed U m s⁻¹ from A in the direction OA and moves in a straight line. After projection, when the displacement of P from O is x m, the velocity of P is v m s⁻¹. The coefficient of friction between the surface and P is 0.4. A force of magnitude 0.8/x N acts on P in the direction PO.
- (i)Show that, while the particle is in motion, v dv/dx = -4 - 2/x.[3]
- (ii)It is given that P comes to instantaneous rest between x = 2.0 and x = 2.1. Find the set of possible values of U.[5]
Mark scheme
Marking points from the official mark scheme (Feb/Mar 2017). The mark or mark type is shown in brackets.
- [B1] Uses F = μR (Friction = 0.4 x 0.4g)
- [M1] Uses Newton's Second Law and a = vdv/dx (0.4vdv/dx = -0.4 x 0.4g – 0.8/x)
- [A1] vdv/dx = -4-2/x
- [M1] Separates the variables and attempts to integrate (∫vdv = ∫(-4 - 2/x)dx)
- [A1] v²/2 = -4x – 2lnx (+ c)
- [M1] Attempts to find c (v = U when x = 1 hence c = U²/2 + 4)
- [M1] Put v = 0 and x = 2
- [A1] Put v = 0 and x = 2.1 (3.28 < U < 3.43)
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