Problems

Two equal smooth spheres of mass \(m\), perfectly elastic, collide. Initially one is at rest. Prove that the velocities of the two spheres after impact are at right angles, and express the final velocities in terms of the initial velocity \(V\) and the angle \(\theta\) between the line of centres at impact and the direction of motion of the moving sphere before impact.

\(O\) is the centre of a regular polygon of \(n\) sides and \(a\) is its distance from each side; \(P\) is a point (inside the polygon) whose polar co-ordinates referred to \(O\) as pole and any initial line are \((b, \alpha)\); \(S_m^{(n)}\) is the sum of the \(m\)th powers of the distances of \(P\) from the sides of the polygon. Shew that \[ S_1^{(n)} = na, \quad S_2^{(n)} = n \left(a^2 + \frac{1}{2}b^2\right). \] It is a known theorem that \(S_3^{(n)}\) is not (like \(S_1^{(n)}, S_2^{(n)}\)) independent of \(\alpha\); verify this when \(n=3\).

From \(H\) a fixed point on a parabola chords \(HP\), \(HQ\) are drawn perpendicular to each other. Shew that the locus of the intersection of tangents to the parabola at \(P\) and \(Q\) is a straight line.

A particle moves in a straight line \(OA\), starting from rest at \(A\), under the action of a force directed always towards \(O\) and proportional to the distance from \(O\); investigate the motion, finding the position and the velocity at any subsequent time. Investigate the motion of the particle if the point \(O\) be not fixed but move with uniform acceleration. Investigate also the motion of a particle moving in a plane under the action of a force directed always to a point \(O\) fixed in the plane and proportional to the distance from \(O\).

Shew that a motor-car, for which the retarding force at \(V\) miles an hour when the brakes are acting may be expressed as \((1000+0.08V^2)\) pounds weight per ton of car, can be stopped in approximately 57 yards from a speed of 50 miles an hour. [\(\log_e 10 = 2.30\)]

\(A, B, C\) are three points in order on a straight line; the segments \(AB, BC\) subtend angles \(\alpha, \beta\) respectively at a point \(P\), and \(\beta + \delta, \alpha - \delta\) respectively at a point \(Q\), \(PQ\) being parallel to \(AC\). Find expressions for the ratios \(BA:BC:BP^2:BQ^2\).

Find the equations of the four straight lines other than the axes which are normal to both of the ellipses \begin{align*} \frac{x^2}{a^2} + \frac{y^2}{b^2} &= 1, \\ \frac{x^2}{b^2} + \frac{y^2}{a^2} &= 1. \end{align*}

A particle moves in a plane under the action of a given system of forces; establish the `principle of work,' namely, that the increase in the kinetic energy of the particle, during any time, is equal to the work done by the forces during that time. Explain clearly under what circumstances the particle may be said to possess a `potential energy,' and shew that, in this case, the `principle of work' becomes the `principle of conservation of energy.' Examine the conception of `angular momentum' and find an expression for the rate of change of angular momentum of the particle, about a fixed point in the plane of motion. A system consists of two particles of masses \(m_1\) and \(m_2\) and velocities \(v_1\) and \(v_2\). Shew that the kinetic energy of the system is equal to the kinetic energy of the motion relative to the centre of gravity of the system together with the kinetic energy of a mass \(m_1+m_2\) moving with the velocity of the centre of gravity. Express the angular momentum of the system about any fixed point in terms of the angular momentum relative to the centre of gravity and the velocity of the centre of gravity.

A spring balance consists of a horizontal disc of mass 4 oz.\ carried on a light vertical spring which is compressed \(\frac{1}{2}\) inch by the weight of the disc. An inelastic mass of 8 oz.\ is dropped from a height of 2 inches on to the disc: find its velocity when it leaves the disc in the subsequent ascent.

A rectangular sheet of paper \(OACB\) is folded over so that the corner \(O\) just reaches a point \(P\) on one of the sides \(AC, CB\). Shew that, as \(P\) moves from \(A\) through \(C\) to \(B\), the length of the crease can pass through a maximum only when it ends at one or other of the corners \(A, B\). Shew also that, if \(a^2/b^2 < \frac{1}{2}(\sqrt{5}-1)\) (where \(OA=a, OB=b\)) the crease will be longest when it passes through \(B\), but, if \(\frac{1}{2}(\sqrt{5}-1) < a^2/b^2 < 1\), it will be longest when it passes through \(A\).