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2016 Paper 3 Q11
D: 1700.0 B: 1484.0
variable force power resistance Newton's second law integration logarithm partial fractions acceleration car motion inequality comparison

A car of mass \(m\) travels along a straight horizontal road with its engine working at a constant rate \(P\). The resistance to its motion is such that the acceleration of the car is zero when it is moving with speed \(4U\).

  1. Given that the resistance is proportional to the car's speed, show that the distance \(X_1\) travelled by the car while it accelerates from speed \(U\) to speed \(2U\), is given by \[ \lambda X_1 = 2\ln \tfrac 9 5 - 1 \,, \] where \(\lambda= P/(16mU^3)\).
  2. Given instead that the resistance is proportional to the square of the car's speed, show that the distance \(X_2\) travelled by the car while it accelerates from speed \(U\) to speed \(2U\) is given by \[ \lambda X_2 = \tfrac43 \ln \tfrac 98 \,. \]
  3. Given that \(3.17<\ln 24 < 3.18\) and \(1.60<\ln 5 < 1.61\), determine which is the larger of \(X_1\) and \(X_2\).

Solution:

  1. \(\,\) \begin{align*} && F_{res} &= kv \\ && P &= Fv \\ v = 4U: && 0 &= F-F_{res} \\ \Rightarrow && 0 &= \frac{P}{4U} - 4Uk \\ \Rightarrow && k &= \frac{P}{16U^2} \\ \\ &&m v \frac{\d v}{\d x}&= \frac{P}{v} - \frac{P}{16U^2}v \\ \Rightarrow && X_1 &= \int_{v=U}^{v=2U} \frac{16U^2mv^2}{P(16U^2-v^2)} \d v \\ v = Ut&& &= \frac{16mU^2}{P} \int_{t=1}^{t=2}\left ( \frac{t^2}{16-t^2} \right)U\d t \\ &&&= \frac{16mU^3}{P} \int_1^2 \left ( -1 + \frac{16}{16-t^2} \right) \d t \\ &&&= \frac{16mU^3}{P} \int_1^2 \left ( -1 +\frac{2}{4+t} +\frac{2}{4-t} \right) \d t \\ &&&= \frac{1}{\lambda}\left (-1 + 2\ln(6)-2\ln(2)-2\ln(5)+2\ln(3) \right) \\ \Rightarrow && \lambda X_1 &= 2\ln \tfrac95-1 \end{align*}
  2. \(\,\) \begin{align*} && F_{res} = kv^2 \\ v = 4U: && 0 &= \frac{P}{4U} - 16U^2k \\ \Rightarrow && k &= \frac{P}{64U^3} \\ \\ && mv \frac{\d v}{\d x} &= \frac{P}{v} - \frac{P}{64U^3}v^2 \\ \Rightarrow && X_2 &= \int_{v=U}^{v=2U} \frac{64U^3mv^2}{P(64U^3-v^3)} \d v \\ &&&= \frac{64U^3m}{P} \int_{v=U}^{v=2U} \frac{v^2}{64U^3-v^3} \d v\\ v = Ut &&&= \frac{64U^3m}{P} \int_{t=1}^{t=2} \frac{U^2t^2}{64U^3-U^3v^3} U \d t\\ &&&= \frac{4}{\lambda} \int_1^2 \frac{t^2}{64-t^3} \d t \\ &&&= \frac{4}{\lambda} \left [ -\frac13\ln(64-t^3) \right]_1^2 \\ &&&= \frac{4}{3\lambda} \ln (63/56) \\ \Rightarrow && \lambda X_2 &= \tfrac43 \ln \tfrac98 \end{align*}
  3. \(\,\) \begin{align*} && 2\ln \tfrac95 - 1 &\overset{?}{>} \frac43 \ln \frac98 \\ \Leftrightarrow && 4 \ln 3 - 2\ln 5 - 1 &\overset{?}{>} \frac83\ln 3 -4 \ln 2 \\ \Leftrightarrow && \frac43(3\ln 3 + 3\ln 2 - 2 \ln 3) &\overset{?}{>} 2 \ln 5 + 1\\ \Leftrightarrow && \frac43\ln 24 &\overset{?}{>} 2 \ln 5 + 1\\ \end{align*} The \(LHS\) is \(>4.22\). The \(RHS\) is \(< 4.22\), and therefore our inequality holds, in particular, \(X_1 > X_2\).

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2009 Paper 2 Q11
D: 1600.0 B: 1516.0
work-energy-power Newton's second law power resistance train logarithmic approximation work-energy theorem kinematics integration optimisation

A train consists of an engine and \(n\) trucks. It is travelling along a straight horizontal section of track. The mass of the engine and of each truck is \(M\). The resistance to motion of the engine and of each truck is \(R\), which is constant. The maximum power at which the engine can work is \(P\). Obtain an expression for the acceleration of the train when its speed is \(v\) and the engine is working at maximum power. The train starts from rest with the engine working at maximum power. Obtain an expression for the time \(T\) taken to reach a given speed \(V\), and show that this speed is only achievable if \[ P>(n+1)RV\,. \]

  1. In the case when \((n+1) RV/P\) is small, use the approximation \(\ln (1-x) \approx -x -\frac12 x^2\) (valid for small \( x \)) to obtain the approximation \[ PT\approx \tfrac 12 (n+1) MV^2\, \] and interpret this result.
  2. In the general case, the distance moved from rest in time \(T\) is \(X\). {\em Write down}, with explanation, an equation relating \(P\), \(T\), \(X\), \(M\), \(V\), \(R\) and \(n\) and hence show that \[ X= \frac{2PT - (n+1)MV^2}{2(n+1)R} \,. \]

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2004 Paper 2 Q11
D: 1600.0 B: 1500.0
variable force power resistance Newton's second law differential equation separable ODE exponential train terminal velocity

The maximum power that can be developed by the engine of train \(A\), of mass \(m\), when travelling at speed \(v\) is \(Pv^{3/2}\,\), where \(P\) is a constant. The maximum power that can be developed by the engine of train \(B\), of mass \(2m\), when travelling at speed \(v\) is \(2Pv^{3/2}.\) For both \(A\) and \(B\) resistance to motion is equal to \(kv\), where \(k\) is a constant. For \(t\le0\), the engines are crawling along at very low equal speeds. At \(t = 0\,\), both drivers switch on full power and at time \(t\) the speeds of \(A\) and \(B\) are \(v_{\vphantom{\dot A}\!A}\) and \(v_{\vphantom{\dot B}\hspace{-1pt}B},\) respectively.

  1. Show that \[ v_{\vphantom{\dot A}\!A} = \frac{P^2 \left(1-\e^{-kt/2m}\right)^2}{k^2} \] and write down the corresponding result for \(v_{\vphantom{\dot B}B}\).
  2. Find \(v_{\vphantom{\dot B}A}\) and \(v_{\vphantom{\dot B}B}\) when \(9 v_{\vphantom{\dot B}A} =4v_{\vphantom{\dot B}B}\;\). [Not on original paper] Show that \(1 < v_{\vphantom{\dot B}\hspace{-1pt}B} /v_{\vphantom{\dot A}\!A} < 4\) for \(t > 0\,\).
  3. Both engines are switched off when \(9 v_{\vphantom{\dot B}A} =4v_{\vphantom{\dot B}B}\,\). Show that thereafter \(k^2 v_{\vphantom{\dot B}B}^2 = 4 P^2 v_{\vphantom{\dot B}A}\,\).

Solution:

  1. \(\,\) \begin{align*} && P &= Fv \\ \text{N2}(\rightarrow): && Pv^{1/2} - kv &= ma \\ \Rightarrow && \dot{v} &= \frac{P}{m} \sqrt{v}-\frac{k}{m}v \\ \Rightarrow && \int \d t & = \int \frac{m}{\sqrt{v}\left (P-k\sqrt{v} \right)} \d v \\ &&t &= -\frac{2m}k\ln\left (P - k\sqrt{v_A}\right) + C \\ t = 0, v_A \approx 0: && 0 & =-\frac{2m}{k} \ln P+ C \\ \Rightarrow && C &= \frac{2m}{k} \ln P \\ \Rightarrow && e^{-kt/2m} &= \frac{P- k \sqrt{v_A}}{P} \\ \Rightarrow && v_A &= \frac{P^2(1-e^{-kt/2m})^2}{k^2} \end{align*} The equation of motion for \(B\) is \(\dot{v_B} = \frac{P}{m}\sqrt{v} - \frac{k}{2m} v\), ie \(k \to \frac{k}{2}\), so \[ v_B = \frac{4P^2(1-e^{-kt/4m})^2}{k^2} \]
  2. Suppose \(9v_A = 4v_B\), then and let \(e^{-kt/4m} = X\) \begin{align*} && 9 \frac{P^2(1-e^{-kt/2m})^2}{k^2} &= 4 \frac{4P^2(1-e^{-kt/4m})^2}{k^2} \\ \Rightarrow && \frac{3}{4} &= \frac{1-X}{1-X^2} \\ \Rightarrow && 0 &= 3X^2-4X+1 \\ &&&= (3X-1)(X-1) \\ \Rightarrow && X &= 1, \frac13 \\ X = 1: && t &= 0 \\ X = \frac13: && e^{-kt/4m} &= \frac13\\ \Rightarrow && t &= \frac{4m}{k}\ln 3 \\ && v_A &= \frac{P^2(1-\frac19)^2}{k^2} \\ &&&= \frac{64P^2}{81k^2} \\ && v_B &= \frac{P^2(1-\frac13)^2}{k^2} \\ &&&= \frac{4P^2}{9k^2} \end{align*} Notice also that \begin{align*} && \frac{v_B}{v_A} &= 4\left ( \frac{1-X}{1-X^2} \right)^2 \\ &&&= 4 \frac{1}{(1+X)^2} \end{align*} Since \(X \in (0,1)\) \(\frac{v_B}{v_A} \in (1, 4)\)
  3. Once the engines are switched off, the equation of motion for \(A\) is (where \(t\) is measured from that point) \begin{align*} && \dot{v} &= -\frac{k}{m}v \\ \Rightarrow && v &= Ae^{-kt/m} \\ \Rightarrow && v_A &= \frac{64P^2}{81k^2}e^{-kt/m} \end{align*} Similarly, \(v_B = \frac{4P^2}{9k^2}e^{-kt/2m}\) so \begin{align*} && \frac{v_A}{v_B^2} &= \frac{64P^2}{81k^2} \cdot \frac{81k^4}{16P^4} = \frac{4k^2}{P^2} \end{align*} as required.

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