M3 June 2005 Q7

EdexcelOld spec14 marksVariable Force & Kinematics

7. A particle \(P\) of mass \(\tfrac{1}{3}\) kg moves along the positive \(x\)-axis under the action of a single force. The force is directed towards the origin \(O\) and has magnitude \(\dfrac{k}{(x+1)^2}\) N, where \(OP = x\) metres and \(k\) is a constant. Initially \(P\) is moving away from \(O\). At \(x = 1\) the speed of \(P\) is 4 m s\(^{-1}\), and at \(x = 8\) the speed of \(P\) is \(\sqrt{2}\) m s\(^{-1}\).

(a) Find the value of \(k\). (10)

(b) Find the distance of \(P\) from \(O\) when \(P\) first comes to instantaneous rest. (4)

Mark scheme (a)

Scheme	Marks
\(\dfrac{1}{3}\ddot{x} = -\dfrac{k}{(x+1)^2}\)	M1
\(\dfrac{1}{3}v\dfrac{\mathrm{d}v}{\mathrm{d}x} = -\dfrac{k}{(x+1)^2}\)	M1
\(\displaystyle\int v\,\mathrm{d}v = \int -\frac{3k}{(x+1)^2}\,\mathrm{d}x\)
\(\dfrac{1}{2}v^2 = \dfrac{3k}{x+1}\ \ (+C)\) Separating variables & attempting integration of both sides	M1 A1=A1
\(v^2 = \dfrac{6k}{x+1} + A\)
Using boundary values to obtain two simultaneous equations.	M1
\((1, 4)\) \(16 = 3k + A\)	A1
\((8, \sqrt{2})\) \(2 = \dfrac{2k}{3} + A\)	A1
\(14 = \dfrac{7}{3}k \Rightarrow k = 6\)	M1 A1
	(10)

Scheme	Marks
\(A = -2\)	B1
\(v^2 = \dfrac{36}{x+1} - 2 = 0\)	M1
\(x = 17\) (m)	M1 A1
	(4)
	(14 marks)

Notes