Show that in rectilinear motion the time taken for any change of velocity is given by the area of the curve connecting the reciprocal of the acceleration and the corresponding velocity. A tramcar starts from rest with an acceleration of 3 ft. per sec. per sec.; the relation between acceleration and speed is linear and the acceleration is 1 ft. per sec. per sec. when the speed is 5 miles an hour. Prove graphically or otherwise that the time taken to reach this speed is 4\(\cdot\)03 seconds. [\(\log_{10} e = \cdot 4343\).]
Develop the theory of the motion of projectiles under gravity, finding the focus, directrix and equation of the path, the maximum range on an inclined plane and the necessary conditions of projection for the projectile to pass through a given point. A gun is firing in the vertical plane passing through the well-defined summit of a hill. Give a diagram shewing the region of space within which lie all points which can be reached by the projectile.
Prove that an ellipse of eccentricity \(1/\sqrt{2}\) will cut at right angles every parabola described with vertex at the centre of the ellipse and axis along the minor axis of the ellipse.
Prove Leibnitz' formula for the \(n\)th differential coefficient of a product of two functions. \(y\) is a solution of the equation \[ 4x(1-x)\frac{d^2y}{dx^2} + 2(1-3x)\frac{dy}{dx}-y = 2+\log(1-x) \] for which \(\frac{dy}{dx}\) is finite when \(x=0\). Prove, by induction or otherwise, that \[ \left(\frac{d^n y}{dx^n}\right)_{x=0} = n!\left(\frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n} + \frac{1}{2n+1}\right). \]
If \(P\) and \(Q\) are two points on the trajectory of a projectile at which the inclinations to the horizontal are \(\alpha\) and \(\theta\), prove that \(\tan\alpha + \tan\theta = 2\tan\gamma\), where \(\gamma\) is the angle of elevation of \(Q\) as seen from \(P\). Show that, if \(\alpha\) and \(\beta\) are the inclinations at \(P\) of the two paths with the same velocity of projection, \(\theta\) and \(\phi\) the inclinations at \(Q\), \(\alpha + \beta - \theta - \phi = \pi\).
Prove the principles of Conservation of Momentum and of Kinetic Energy for a material system. A particle of mass \(m\) is placed at the vertex of a hemisphere of mass \(M\), whose base rests on a smooth horizontal plane, and is slightly disturbed. Shew that the path in space of the particle is an arc of an ellipse, as long as contact is preserved. Shew that contact ceases when the radius to the particle makes an angle \(\theta\) with the vertical given by the equation \[ m \cos^3\theta - 3(M+m)\cos\theta + 2(M+m) = 0; \] and verify that if \(M = m \tan^2\alpha\) (\(\alpha < \pi/2\)), the value to be taken for \(\cos\theta\) is \[ 2 \sec \alpha \cos \frac{\pi+\alpha}{3}. \]
Differentiate \[ \tan^{-1} \frac{1+x}{1-x}, \quad \log (\tan x + \sec x). \] Find the \(n\)th differential coefficients of \(\sin x\) and \(\sin^3 x\).
A curve of degree three is represented by the equation \(\phi(x,y)=0\) in which the coefficients are rational numbers. Prove that the tangent at any point whose coordinates \((x_n, y_n)\) are rational numbers meets the curve again in a point \((x_{n+1}, y_{n+1})\) whose coordinates are rational. If the curve is \[ x^3+y^3=9, \] find the relation between \(x_{n+1}\) and \(x_n\).
A wheel, which can rotate in a vertical plane about a horizontal axis through its centre, carries a particle at its rim, and motion is resisted by a constant frictional couple, such that in limiting equilibrium the radius to the particle makes 30\(^\circ\) with the vertical. Show that, if the wheel is slightly disturbed from the position of unstable equilibrium, the angle described in the first swing is between 188\(^\circ\) and 189\(^\circ\), and that in the second swing the particle comes to rest before the radius to it reaches the vertical.
A man is 2 miles from the nearest point \(A\) of a straight road, and he wishes to reach a point \(B\) on the road 4 miles from \(A\). He can walk at 4 miles per hour until he reaches the road and at 5 miles per hour on the road. Find the least time in which he can reach \(B\).