A-LevelMathematicsDifferential equations9709/52 · Oct/Nov 2019 · Q69 Marks
A particle P of mass 0.2kg is projected horizontally from a fixed point O on a smooth horizontal surface. When the displacement of P from O is xm the velocity of Pis vms¯¹. A horizontal force of variable magnitude 0.09√x N directed away from O acts on P. An additional force of constant magnitude 0.3 N directed towards O acts on P.
- (i)Show that v dv/dx = 0.45√x - 1.5.[2]
- (ii)Find the value of x for which the acceleration of P is zero.[2]
- (iii)Given that the minimum value of v is positive, find the set of possible values for the speed of projection.[5]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.
- [M1] 0.2v dv/dx = 0.09√x - 0.3
- [A1] v dv/dx = 0.45√x - 1.5
- [M1] 0 = 0.45x^(1/2) - 1.5
- [A1] x = 100/9
- [M1] ∫v dv = ∫(0.45x^(1/2) - 1.5)dx
- [A1] v²/2 = 0.3x^(3/2) - 1.5x (+c)
- [M1] 0.3(100/9)^(3/2) - 1.5(100/9) + c = 0
- [A1] c = 50/9
- [A1] x=0, v² > 50/9 so v > 10/3
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