Problems

Two smooth elastic spheres of equal mass are moving in the same direction in parallel paths with velocities \(u, u'\). Prove that, if the spheres impinge, the greatest deviation between their paths after impact is \[ \tan^{-1}\left[\frac{1}{2}(1+e)\left(\frac{u'}{u}-\frac{u}{u'}\right) / \left\{2(1+e^2) + (1-e)^2\left(\frac{u'}{u}+\frac{u}{u'}\right)\right\}\right]. \]

A large number of cards, which are of \(r\) different kinds, are contained in a box from which a man draws one \(n\) times successively. On each occasion it is equally likely that the card drawn will be of any particular kind. Prove that the probability that after drawing \(n\) cards the man will have a complete set, i.e. at least one of every kind, is the sum of the coefficients of all terms in the expansion of \[ (x_1+x_2+x_3+\dots+x_r)^n \] which contain every one of the \(r\) quantities \(x\), divided by \(r^n\).

The internal and external bisectors of the angle \(A\) of the triangle \(ABC\) are drawn meeting \(BC\) in \(D\) and \(D'\). Find the lengths of \(AD\) and \(AD'\) and prove that \[ AD^2+AD'^2 = \frac{4a^2b^2c^2}{(b^2-c^2)^2}. \]

A particle moving in vacuo passes with a given velocity \(q\) through a fixed point \(O\). Shew that all possible paths are parabolas of which the directrices lie in a common horizontal plane; and use this result to determine, by means of a geometrical construction, the paths which contain a second specified point. If the path also passes through \(Q\), where \(OQ\) makes a given angle \(\alpha\) with the horizontal, determine the conditions under which the distance \(OQ\) will have its maximum value, and prove that, when these conditions are satisfied:

1. [(i)] the directions of motion at \(O\) and at \(Q\) are perpendicular to one another;
2. [(ii)] the time occupied in passing from \(O\) to \(Q\) is equal to the time which would be occupied in dropping from rest through a vertical distance equal to \(OQ\);
3. [(iii)] \(Q\) lies on a fixed paraboloid having its focus at \(O\).

A heavy particle is attached to the rim of a wheel of radius \(r\) which is made to rotate in a vertical plane with constant angular velocity \(\omega\) about its centre which is fixed. Shew that if the particle is set free at some point of its path the time \(t\) taken to reach the horizontal plane through the lowest point of the wheel is given by \(g^2 t^4 - 2 t^2 r (g+r\omega^2) + x^2 = 0\), where \(x\) is the horizontal distance traversed measured from the lowest point of the wheel. Deduce that the greatest value of \(x\) is \(r + \omega^2 r^2 / g\).

Prove that, if \(x\) and \(y\) are real, \[ |\cot(x+iy)| < |\coth y|, \quad |\tan(x+iy)| < |\coth y|, \] where \(|a+ib|\) denotes as usual \(+\sqrt{(a^2+b^2)}\).

Find the equation of the normal at the point \(P(am^2, 2am)\) of the parabola \(y^2 = 4ax\) and the co-ordinates of the point \(Q\) in which it meets the curve again. If \(\phi\) is the angle between the tangents to the parabola at \(P\) and \(Q\) shew that \[ \tan \phi = 2 \tan\theta, \] where \(\theta\) is the angle which the tangent at \(P\) makes with the axis of the parabola.

Prove that in the motion of a system of particles in one plane:

1. [(i)] the increase of kinetic energy in any time is equal to the work done by the forces in that time;
2. [(ii)] the centre of mass moves as a particle of mass equal to that of the system under the action of all the forces moved parallel to themselves to act at the centre of mass;
3. [(iii)] the motion relative to the centre of mass is independent of the motion of that point.

A uniform rod of length \(l\) has one end fastened to a fixed point which is at a height \(l\) above the ground. The rod is given an initial angular velocity \(\omega\) in a vertical plane, starting from rest in its lowest position; and the fastening gives way when the rod first reaches a horizontal position. Shew that, if the rod is horizontal when it strikes the ground, then \[ l\omega^2 = 3g + gk^2\pi^2/(2+k\pi), \] where \(k\) is a positive integer.

At speeds over 8 miles an hour, the total tractive force at the rims of the wheels of an 11 ton tramcar is given by the equation \(P(v-5) = 7000\), where \(P\) is the force in pounds weight and \(v\) is the velocity in miles an hour. Shew that the tramcar can accelerate from 8 to 12 miles an hour in about 16 yards.

Express \[ \left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial u}{\partial y}\right)^2 + \left(\frac{\partial u}{\partial z}\right)^2 \] in terms of \(r, \theta, \phi\), and the partial derivatives of \(u\) with respect to \(r, \theta, \phi\), where \[ x = r \sin\theta \cos\phi, \quad y = r \sin\theta \sin\phi, \quad z=r\cos\theta. \]