A car has two gears, and its performance (after allowing for air resistance and friction) is such that in bottom gear the acceleration is \(2C(V-v)\), and in top gear it is \(C(V-\frac{1}{2}v)\). Here \(v\) is the speed of the car, \(C\) and \(V\) are constants. The car is started from rest and accelerated as quickly as possible, the gear change occupying negligible time. Show that a speed \(v = 4V/3\) is reached after time \(C^{-1}\log 4/3\), and calculate the distance travelled by then.
A plane lamina is acted on by forces having components \((X_r, Y_r)\) at points \((x_r, y_r)\) \((r = 1, 2, \ldots)\), in Cartesian coordinates. Writing \(z_r = x_r + iy_r\) and \(Z_r = X_r + iY_r\) (so that the points and forces may be represented in the complex plane), write down the complex number representing the resultant force, and show that the moment of the system about the point \((a, b)\) is $$-\mathcal{I}\sum_r[(z_r-c)\bar{Z_r}],$$ where \(c = a + ib\), the bar denotes the complex conjugate, and \(\mathcal{I}\) denotes the imaginary part. Using these formulae, or otherwise, show that if all the forces are turned through the same angle in the same sense, their resultant always passes through a fixed point, whose Cartesian coordinates should be obtained. [Assume that the resultant force does not vanish.]
A quadrilateral \(ABCD\) is formed from four uniform rods freely jointed at their ends. The rods \(AB\) and \(DA\) are equal in length and weight, and so also are the rods \(BC\) and \(CD\). The quadrilateral is suspended from \(A\) and a string joins \(A\) and \(C\) so that \(ABC\) is a right angle and the angle \(BAD = 2\theta\). Show that the tension in the string is \(w' + (w + w')\sin^2\theta\), where \(w\) is the weight of \(AB\) and \(w'\) is the weight of \(BC\).
Assuming that Oxford and Cambridge are 65 miles apart, and are at the same height above sea level, show that if a straight tunnel were bored between them, a train would traverse it (in one direction) under gravity alone in about 42 minutes, and find the maximum speed attained. Would your results be substantially modified if the tunnel were bored between Land's End and John O'Groats (about 600 miles apart)?
An ammeter has a coil and needle which rotate on a pivot. The moving system has moment of inertia \(M\), and when the deflection is \(\theta\), a spring supplies a restoring couple \(-A\theta\). When a current \(I\) flows the coil experiences a deflecting couple \(kI\). There is also a resisting couple \(-\mu\dot{\theta}\) due to friction and eddy currents whenever the coil is moving. Here \(M\), \(k\), \(\lambda\) and \(\mu\) are all constants. Show that the current and deflection are related by $$M\ddot{\theta} + \mu\dot{\theta} + A\theta = kI.$$ Find the complementary function for this equation, distinguishing between the cases where \(\mu^2\) is greater than, less than, and equal to \(4MA\). Explain how to solve the problem in which \(\theta\), \(\dot{\theta}\) and \(I\) are initially zero, and a steady current \(I\) is switched on at a time \(t = 0\). Why is the choice \(\mu^2 = 4MA\) the most convenient in practice?
A uniform rod \(AB\), of length \(a\) and mass \(m\), is pivoted about \(A\). It is released from rest with \(B\) vertically above \(A\), and given a very slight displacement so that it falls under gravity. Find the horizontal and vertical components of the reaction at the pivot when the rod makes an angle \(\theta\) with the upwards vertical.
A rocket is travelling horizontally. Its initial mass is \(M\) and it expels a mass \(m\) of gas per unit time horizontally with a velocity \(a\) relative to the rocket, where \(m\) and \(a\) are constants. If the rocket experiences a resistive force which is a constant multiple \(k\) of its velocity \(v\), show that if \(v = 0\) when \(t = 0\) $$\left(\frac{M-mt}{M}\right)^k = \left(\frac{ma-kv}{ma}\right)^m.$$ Find a similar relation for the case where the resistive force is proportional to the square of the velocity of the rocket.
A plane \(P\) passing through a point \(O\) is inclined at \(30^\circ\) to the horizontal. A ball, whose coefficient of restitution with \(P\) is \(\frac{1}{3}\), is projected from \(O\) in a vertical plane through the line of greatest slope of \(P\) with speed \(V\), at an angle of \(60^\circ\) to the horizontal and \(30^\circ\) to the line of greatest slope. Find the maximum height above \(O\) (measured vertically) that it attains between the first and second bounces.
A plane lamina in the shape of a quadrant of the unit circle has a variable density proportional to \(r^{-1}\sin(\frac{1}{2}\pi r)\) where \(r\) is the distance from the centre of the circle. Calculate its moment of inertia about an axis through its centre of gravity perpendicular to the plane of the lamina.
Two numbers \(X\) and \(Y\) between 1 and 100 (inclusive) are selected at random, all possible pairs \((X, Y)\) having equal probabilities. Let \(Z\) denote the maximum of \(X\) and \(Y\). What is the probability that \(Z \leqslant 50\)? By use of the formulae $$\sum_{r=1}^n r = \frac{1}{2}n(n+1),$$ $$\sum_{r=1}^n r^2 = \frac{1}{6}n(n+1)(2n+1),$$ or otherwise, show that the mean of \(Z\) is just over 67. Find a median of \(Z\). [A median of \(Z\) is any number \(\xi\) such that \(P\{Z \leqslant \xi\} \geqslant \frac{1}{2}\) and \(P\{Z \geqslant \xi\} \geqslant \frac{1}{2}\)]