M5 June 2007 Q7

EdexcelOld spec14 marksImpulse & Momentum

7. A motor boat of mass \(M\) is moving in a straight line, with its engine switched off, across a stretch of still water. The boat is moving with speed \(U\) when, at time \(t = 0\), it develops a leak. The water comes in at a constant rate so that at time \(t\), the mass of water in the boat is \(\lambda t\). At time \(t\) the speed of the boat is \(v\) and it experiences a total resistance to motion of magnitude \(2\lambda v\).

(a) Show that \((M + \lambda t)\dfrac{\mathrm{d}v}{\mathrm{d}t} + 3\lambda v = 0\). (6)

(b) Show that the time taken for the speed of the boat to reduce to \(\tfrac{1}{2}U\) is \(\dfrac{M}{\lambda}\left(2^{\frac{1}{3}} - 1\right)\). (6)

The boat sinks when the mass of water inside the boat is \(M\).

Mark scheme (a)

Scheme	Marks
\((m + \delta m)(v + \delta v) - mv = -2\lambda v\,\delta t\)	M1 A1
\(m\dfrac{\mathrm{d}v}{\mathrm{d}t} + v\dfrac{\mathrm{d}m}{\mathrm{d}t} = -2\lambda v\)
\(\dfrac{\mathrm{d}m}{\mathrm{d}t} = \lambda\); \(m = M + \lambda t\)	B1; B1
\((M + \lambda t)\dfrac{\mathrm{d}v}{\mathrm{d}t} + 3\lambda v = 0\) *	DM1 A1
	(6)

Mark scheme (b)

Scheme	Marks
\(-\displaystyle\int\frac{\mathrm{d}v}{3\lambda v} = \int\frac{\mathrm{d}t}{(M + \lambda t)}\)	M1
\(-\dfrac{1}{3\lambda}\Big[\ln v\Big]_U^{\frac{1}{2}U} = \dfrac{1}{\lambda}\Big[\ln(M + \lambda t)\Big]_0^T\)	DM1 A1
\(\dfrac{1}{3}\ln 2 = \ln\dfrac{(M + \lambda T)}{M}\)	DM1
\(T = \dfrac{M}{\lambda}\left(2^{\frac{1}{3}} - 1\right)\) *	DM1 A1
	(6)

Notes

(Corrected from the printed mark scheme: the limits on \(\ln v\) are printed as \(u\) and \(\tfrac{1}{2}u\); the question uses \(U\).)

Mark scheme (c)

Scheme	Marks
Sinks at \(T_S = \dfrac{M}{\lambda}\)
Reaches speed \(\dfrac{1}{2}U\) at \(T = \dfrac{M}{\lambda}\left(2^{\frac{1}{3}} - 1\right)\)
Since \(\left(2^{\frac{1}{3}} - 1\right) \lt 1\), \(T \lt T_S\)	M1
i.e. Reaches speed \(\dfrac{1}{2}U\) before it sinks	A1 c.s.o.
	(2)
	(14 marks)