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2021 Paper 2 Q10
A train moves westwards on a straight horizontal track with constant acceleration \(a\), where \(a > 0\). Axes are chosen as follows: the origin is fixed in the train; the \(x\)-axis is in the direction of the track with the positive \(x\)-axis pointing to the East; and the positive \(y\)-axis points vertically upwards. A smooth wire is fixed in the train. It lies in the \(x\)--\(y\) plane and is bent in the shape given by \(ky = x^2\), where \(k\) is a positive constant. A small bead is threaded onto the wire. Initially, the bead is held at the origin. It is then released.
- Explain why the bead cannot remain stationary relative to the train at the origin.
- Show that, in the subsequent motion, the coordinates \((x, y)\) of the bead satisfy \[ \dot{x}(\ddot{x} - a) + \dot{y}(\ddot{y} + g) = 0 \] and deduce that \(\tfrac{1}{2}(\dot{x}^2 + \dot{y}^2) - ax + gy\) is constant during the motion.
- Find an expression for the maximum vertical displacement, \(b\), of the bead from its initial position in terms of \(a\), \(k\) and \(g\).
- Find the value of \(x\) for which the speed of the bead relative to the train is greatest and give this maximum speed in terms of \(a\), \(k\) and \(g\).
2017 Paper 2 Q10
A car of mass \(m\) makes a journey of distance \(2d\) in a straight line. It experiences air resistance and rolling resistance so that the total resistance to motion when it is moving with speed \(v\) is \(Av^2 +R\), where \(A\) and \(R\) are constants. The car starts from rest and moves with constant acceleration \(a\) for a distance \(d\). Show that the work done by the engine for this half of the journey is \[ \int_0^d (ma+R+Av^2) \, \d x \] and that it can be written in the form \[ \int_0^w \frac {(ma+R+Av^2)v}a\; \d v \,, \] where \(w =\sqrt {2ad\,}\,\). For the second half of the journey, the acceleration of the car is \(-a\).
- In the case \(R>ma\), show that the work done by the engine for the whole journey is \[ 2Aad^2 + 2Rd \,. \]
- In the case \(ma-2Aad< R< ma\), show that at a certain speed the driving force required to maintain the constant acceleration falls to zero. Thereafter, the engine does no work (and the driver applies the brakes to maintain the constant acceleration). Show that the work done by the engine for the whole journey is \[ 2Aad^2 + 2 Rd + \frac{(ma-R)^2}{4Aa} \, .\]
Solution: The force delivered by the engine must be \(ma + R + Av^2\), (so the net force is \(ma\)). Therefore the work done is \(\displaystyle \int_0^d F \d x = \int_0^d (ma + R + Av^2) \d x\) Notice that \(a = v \frac{\d v}{\d x} \Rightarrow \frac{a}{v} = \frac{\d v}{\d x}\) and so \begin{align*} && WD &= \int_0^d (ma + R + Av^2) \d x \\ &&&= \int_{x=0}^{x=d} (ma + R + Av^2) \frac{v}{a} \frac{\d v}{\d x} \d x \\ &&&= \int_{x=0}^{x=d} \frac{ (ma + R + Av^2)v}{a} \d v \\ \end{align*} Also notice that if we move with constant acceleration from rest for a distance \(d\) the final speed is \(v^2 = 2ad \Rightarrow v = \sqrt{2ad}\)
- For the second part of the journey, the engine will be putting out a force of \(-ma+R+Av^2>0\), and the car will have a final speed of \(0\) \begin{align*} WD &= \int_0^{w} \frac{(ma+R+Av^2)v}{a} \d v + \int_w^0 \frac{(-ma+R+Av^2)v}{-a} \d v \\ &= \int_0^w \frac{2(Rv+Av^3)}{a} \d v \\ &= \frac{Rw^2+\frac12Aw^4}{a} \\ &= \frac{R2ad+\frac12A4a^2d^2}{a} \\ &= 2Rd + 2Aad^2 \end{align*}
- If \(ma - 2Aad < R < ma\) then the driving force is still \(-ma+R+Av^2\) which is positive when \(v = \sqrt{2as}\) but negative when \(v = 0\), and therefore at some point in-between the driving force must be \(0\). The engine will stop working when \(-ma+R+Av^2 =0 \Rightarrow v = \sqrt{\frac{ma-R}{A}}\) so \begin{align*} WD &= \int_0^w \frac{(ma+R+Av^2)v}{a} \d v + \int_w^{ \sqrt{\frac{ma-R}{A}}} \frac{(-ma+R+Av^2)v}{-a} \d v \\ &= \int_0^w \frac{2(R+Av^2)v}{a} \d v - \int_0^{\sqrt{\frac{ma-R}{A}}} \frac{(-ma+R+Av^2)v}{a}\d v \\ &= 2Aad^2+2Rd + \frac1a\left (\frac12(R-ma)\frac{ma-R}{A} + \frac{A}{4}\left ( \frac{ma-R}{A}\right)^2 \right) \\ &= 2Aad^2+2Rd - \frac{(ma-R)^2}{Aa}\left (-\frac12+ \frac14 \right) \\ &= 2Aad^2+2Rd - \frac{(ma-R)^2}{4Aa} \end{align*}
2012 Paper 1 Q10
I stand at the top of a vertical well. The depth of the well, from the top to the surface of the water, is \(D\). I drop a stone from the top of the well and measure the time that elapses between the release of the stone and the moment when I hear the splash of the stone entering the water. In order to gauge the depth of the well, I climb a distance \(\delta\) down into the well and drop a stone from my new position. The time until I hear the splash is \(t\) less than the previous time. Show that \[ t = \sqrt{\frac{2D}g} - \sqrt{\frac{2(D-\delta)}g} + \frac \delta u\,, \] where \(u\) is the (constant) speed of sound. Hence show that \[ D = \tfrac12 gT^2\,, \] where \(T= \dfrac12 \beta + \dfrac \delta{\beta g}\) and \(\beta = t - \dfrac \delta u\,\). Taking \(u=300\,\)m\,s\(^{-1}\) and \(g=10\,\)m\,s\(^{-2}\), show that if \(t= \frac 15\,\)s and \(\delta=10\,\)m, the well is approximately \(185\,\)m deep.
Solution: \begin{align*} && s &= ut + \frac12at^2 \\ && D &= \frac12gt_D^2 \\ \Rightarrow && t_D &= \sqrt{\frac{2D}{g}} \\ \Rightarrow && t_{D-\delta} &= \sqrt{\frac{2(D-\delta}{g}} \end{align*} Therefore the difference in times of what I hear will be: \begin{align*} t &= \underbrace{\sqrt{\frac{2D}{g}}}_{\text{time for first stone to hit water}} + \underbrace{\frac{D}{u}}_{\text{time to hear about it}} - \left (\underbrace{\sqrt{\frac{2(D-\delta)}{g}}}_{\text{time for second stone to hit water}} + \underbrace{\frac{D-\delta}{u}}_{\text{time to hear about it}} \right) \\ &= \sqrt{\frac{2D}g} - \sqrt{\frac{2(D-\delta)}g} + \frac \delta u \end{align*} \begin{align*} && t &= \sqrt{\frac{2D}g} - \sqrt{\frac{2(D-\delta)}g} + \frac \delta u \\ \Rightarrow && \beta &= \sqrt{\frac{2D}g} - \sqrt{\frac{2(D-\delta)}g} \\ && \beta^2 &= \frac{2D}{g} + \frac{2(D-\delta)}{g} - \frac{4}{g}\sqrt{D(D-g)} \\ &&&= \frac{4D}{g} - \frac{2\delta}{g} - \frac{4}{g} \sqrt{D(D-\delta)}\\ \Rightarrow && g\beta^2 &= 4D-2\delta -4\sqrt{D(D-\delta)}\\ \Rightarrow && (g \beta^2-4D+2\delta)^2 &= 16D(D-\delta) \\ \Rightarrow && g^2\beta^4 + 16D^2 + 4\delta^2 -8g\beta^2D+4g\beta^2 \delta -16D\delta &= 16D^2-16D\delta \\ \Rightarrow && 8g\beta^2D &= g\beta^4 +4\delta^2 +4g\beta^2 \delta \\ \Rightarrow && D &= \frac1{8g\beta^2} \left ( g^2\beta^4 +4\delta^2 +4g\beta^2 \delta\right) \\ &&&= \frac1{8g\beta^2} \left ( g\beta^2 +2\delta \right)^2 \\ &&&= \frac12g \left ( \frac{\beta}{2} + \frac{\delta}{g\beta} \right)^2 \end{align*} If \(u = 300, g = 10, t = \frac15, \delta = 10\), then \begin{align*} && \beta &= \frac15-\frac{10}{300}\\ &&&= \frac15 - \frac1{30} \\ &&&= \frac{1}{6}\\ && D &= \frac12 \cdot 10 \left ( \frac1{12} + 6 \right)^2 \\ &&&= 5\cdot (36 + 1 + \frac{1}{12^2}) \\ &&&\approx 37 \cdot 5 = 185 \end{align*}
2008 Paper 1 Q10
On the (flat) planet Zog, the acceleration due to gravity is \(g\) up to height \(h\) above the surface and \(g'\) at greater heights. A particle is projected from the surface at speed \(V\) and at an angle \(\alpha\) to the surface, where \(V^2 \sin^2\alpha > 2 gh\,\). Sketch, on the same axes, the trajectories in the cases \(g'=g\) and \(g' < g\). Show that the particle lands a distance \(d\) from the point of projection given by \[ d = \left(\frac {V-V'} g + \frac {V'}{ g'} \right) V\sin2\alpha\,, \] where \(V' = \sqrt{V^2-2gh\,\rm{cosec}^2\alpha\,}\,\).
2007 Paper 1 Q10
The Norman army is advancing with constant speed \(u\) towards the Saxon army, which is at rest. When the armies are \(d\) apart, a Saxon horseman rides from the Saxon army directly towards the Norman army at constant speed \(x\). Simultaneously a Norman horseman rides from the Norman army directly towards the Saxon army at constant speed \(y\), where $y > u$. The horsemen ride their horses so that \(y - 2x < u < 2y - x\). When each horseman reaches the opposing army, he immediately rides straight back to his own army without changing his speed. Represent this information on a displacement-time graph, and show that the two horsemen pass each other at distances \[ \frac{xd }{ x + y} \;\; \mbox{and} \;\; \frac{xd(2y -x-u)} {(u+x ) ( x + y )} \] from the Saxon army. Explain briefly what will happen in the cases (i) \(u > 2y - x\) and (ii) \(u < y - 2x\).
2004 Paper 1 Q10
A particle is travelling in a straight line. It accelerates from its initial velocity \(u\) to velocity \(v\), where \(v > \vert u \vert > 0\,\), travelling a distance \(d_1\) with uniform acceleration of magnitude \(3a\,\). It then comes to rest after travelling a further distance \(d_2\,\) with uniform deceleration of magnitude \(a\,\). Show that
- if \(u>0\) then \(3d_1 < d_2\,\);
- if \(u<0\) then \(d_2 < 3d_1 < 2d_2\,\).
2003 Paper 3 Q11
Point \(B\) is a distance \(d\) due south of point \(A\) on a horizontal plane. Particle \(P\) is at rest at \(B\) at \(t=0\), when it begins to move with constant acceleration \(a\) in a straight line with fixed bearing~\(\beta\,\). Particle \(Q\) is projected from point \(A\) at \(t=0\) and moves in a straight line with constant speed \(v\,\). Show that if the direction of projection of \(Q\) can be chosen so that \(Q\) strikes \(P\), then \[ v^2 \ge ad \l 1 - \cos \beta \r\;. \] Show further that if \(v^2 >ad(1-\cos\beta)\) then the direction of projection of \(Q\) can be chosen so that \(Q\) strikes \(P\) before \(P\) has moved a distance \(d\,\).
2002 Paper 2 Q10
A competitor in a Marathon of \(42 \frac38\) km runs the first \(t\) hours of the race at a constant speed of 13 km h\(^{-1}\) and the remainder at a constant speed of \(14 + 2t/T\) km h\(^{-1}\), where \(T\) hours is her time for the race. Show that the minimum possible value of \(T\) over all possible values of \(t\) is 3. The speed of another competitor decreases linearly with respect to time from 16~km~h\(^{-1}\) at the start of the race. If both of these competitors have a run time of 3 hours, find the maximum distance between them at any stage of the race.
1999 Paper 1 Q9
A tortoise and a hare have a race to the vegetable patch, a distance \(X\) kilometres from the starting post, and back. The tortoise sets off immediately, at a steady \(v\) kilometers per hour. The hare goes to sleep for half an hour and then sets off at a steady speed \(V\) kilometres per hour. The hare overtakes the tortoise half a kilometre from the starting post, and continues on to the vegetable patch, where she has another half an hour's sleep before setting off for the return journey at her previous pace. One and quarter kilometres from the vegetable patch, she passes the tortoise, still plodding gallantly and steadily towards the vegetable patch. Show that \[ V= \frac{10}{4X-9} \] and find \(v\) in terms of \(X\). Find \(X\) if the hare arrives back at the starting post one and a half hours after the start of the race.
Solution: If \(T_1\) is the time they meet after \(\frac12\)km from the starting points and \(T_2\) is the time they meet a second time, then \begin{align*} && \frac12 &= vT_1 \\ &&&= V(T_1-\tfrac12) \\ && X - \frac54 &= vT_2 \\ && X + \frac54 &= V(T_2 - 1) \\ && \frac{T_2}{T_1} &= \frac{4X-5}{2} \\ && X + \frac54 + V &= VT_2 \\ && \frac12 + \frac12 V &= VT_1 \\ \Rightarrow && \frac{T_2}{T_1} &= \frac{4X + 5 + 4V}{2(1+V)} \\ \Rightarrow && \frac{4X-5}{2}&=\frac{4X + 5 + 4V}{2(1+V)} \\ \Rightarrow && V(4X-9) &= 10 \\ \Rightarrow && V &= \frac{10}{4X-9} \\ \\ && T_1 &= \frac{1}{2V} + \frac12 \\ &&&= \frac{4X+1}{20} \\ && v &= \frac{1}{2T_1} \\ &&&= \frac{10}{4X+1} \end{align*} \begin{align*} && 2X &= \frac12 V \\ \Rightarrow && 2X(4X-9) &= 5 \\ \Rightarrow && 0 &= 8X^2-18X-5 \\ &&&= (4X+1)(2X-5) \\ \Rightarrow && X &= -\frac14, \frac52 \end{align*} Since \(X\) is positive, we must have \(X = \frac52\)km
1998 Paper 1 Q11
Hank's Gold Mine has a very long vertical shaft of height \(l\). A light chain of length \(l\) passes over a small smooth light fixed pulley at the top of the shaft. To one end of the chain is attached a bucket \(A\) of negligible mass and to the other a bucket \(B\) of mass \(m\). The system is used to raise ore from the mine as follows. When bucket \(A\) is at the top it is filled with mass \(2m\) of water and bucket \(B\) is filled with mass \(\lambda m\) of ore, where \(0<\lambda<1\). The buckets are then released, so that bucket \(A\) descends and bucket \(B\) ascends. When bucket \(B\) reaches the top both buckets are emptied and released, so that bucket \(B\) descends and bucket \(A\) ascends. The time to fill and empty the buckets is negligible. Find the time taken from the moment bucket \(A\) is released at the top until the first time it reaches the top again. This process goes on for a very long time. Show that, if the greatest amount of ore is to be raised in that time, then \(\lambda\) must satisfy the condition \(\mathrm{f}'(\lambda)=0\) where \[\mathrm{f}(\lambda)=\frac{\lambda(1-\lambda)^{1/2}} {(1-\lambda)^{1/2}+(3+\lambda)^{1/2}}.\]
1992 Paper 3 Q11
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