Two identical small smooth spheres \(S_1\) and \(S_2\) of radius \(b\) are free to slide inside a long smooth hollow tube whose inner circular cross-section is just large enough to contain the spheres. The tube has length \(2\pi a\) where \(a\) is much greater than \(b\), and it is bent in the form of a large circle of radius \(a\) and closed on itself. Suppose that the tube is held fixed in a horizontal plane, that \(S_1\) and \(S_2\) are initially touching each other, that \(S_1\) is at rest and that \(S_2\) is projected away from \(S_1\) with speed \(U\). Find, in terms of \(U\), \(a\), and the coefficient of restitution \(e\) for collision between the spheres,
For the purpose of this question it may be assumed that, when any car travelling at speed \(v\) on a straight road makes an emergency stop, it stops in a distance \(t_0 v + bv^n\), where \(t_0\) is the reaction time of the driver (the same for all drivers) and \(b\) and \(n\) are positive constants (the same for all cars) determined by the efficiency of the brakes. A car \(C_1\) travelling at speed \(v_1\) is following a car \(C_2\) travelling at speed \(v_2\) (\(< v_1\)). When the cars are separated by a distance \(d\), the driver of \(C_2\) detects a hazard ahead and makes an emergency stop. When the driver of \(C_1\) sees the brake-lights of \(C_2\) (which light up after the first reaction time \(t_0\)) he also makes an emergency stop. Show that a collision is inevitable if \(v_1 > \lambda v_2\), where \[\lambda^n + \epsilon\lambda = A\] and \(\epsilon\) and \(A\) are to be found, as functions of \(t_0\), \(b\), \(n\), \(d\) and \(v_2\). When \(\epsilon\) is small the solution of this equation may be found as a power series \[\lambda = \lambda_0 + \epsilon\lambda_1 + \epsilon^2\lambda_2 + \ldots\] By substituting, and equating coefficients of different powers of \(\epsilon\), find \(\lambda_0\), \(\lambda_1\) and \(\lambda_2\).
One end \(A\) of a uniform rod \(AB\) of length \(2a\) and weight \(W\) can turn freely about a fixed smooth hinge; the other end \(B\) is attached by a light elastic string of unstretched length \(a\) to a fixed support at the point vertically above \(A\) and distant \(4a\) from \(A\). If the equilibrium of the vertical position of the rod with \(B\) above \(A\) is stable, find the minimum modulus of elasticity of the string.
Serving a ball in the game of lawn tennis can be modelled by the following problem. A projectile is emitted with horizontal and vertical components of velocity \(u\) and \(v\) (\(u > 0\)) from a point at a height \(h\) above horizontal ground. The player can adjust \(u\) and \(v\) but not \(h\). At a horizontal distance \(a\) from the point of emission, there is a net of height \(c\) which the ball must clear; the ball must also strike the ground at a horizontal distance not greater than \(a+b\) from the point of emission. Establish two inequalities involving the quantities \(\xi = u^2\) and \(\eta = uv\) linearly, and show diagrammatically, using the \((\xi, \eta)\) plane, what is required for a valid serve, distinguishing between the cases where \(h\) is less than or greater than \(c(1 + a/b)\).
(i) A smooth tube \(AB\) of length \(\frac{1}{2}\pi a\) and of small cross-section is bent in the form of a circular arc of radius \(a\) and is fixed in a vertical plane with the end \(A\) uppermost, the tangent to the axis of the tube at \(A\) being horizontal and the tangent at \(B\) being vertical. The tube contains a uniform flexible inextensible chain of length \(\frac{1}{2}\pi a\) which just extends throughout the length of the tube. The chain is released from rest and slides out of the tube under gravity. Find its speed when it is just clear of the lower end of the tube. (ii) The same tube is now held fixed in a horizontal plane and a stream of identical particles is fired into the tube at \(A\), emerging from the tube at \(B\). Assuming that each particle loses half its energy in its passage through the tube, and that there is no accumulation of particles in the tube, find the magnitude and direction of the force experienced by the tube in terms of the mass \(m\) of each particle, the number \(N\) per unit time entering the tube at \(A\), and the speed \(v\) of the particles at \(A\). [You are not asked to find the torque or the line of action of the force.]
Let \(\alpha\), \(\beta\), \(\gamma\) be real constants and \(\mathbf{a}\) a real vector in three dimensions. Show that if the equations \[\alpha\mathbf{x}+\beta\mathbf{y} = \mathbf{a},\] \[\mathbf{x}.\mathbf{y}=\gamma\] have real solutions for the vectors \(\mathbf{x}\), \(\mathbf{y}\), then \[\mathbf{a} \cdot \mathbf{a} \geq 4\alpha\beta\gamma.\] Find the general solution of the equations when this inequality is satisfied.