Prove that if two particles of masses \(m_1, m_2\) are moving in a plane, their kinetic energy is \[ \frac{1}{2}(m_1+m_2)V^2+\frac{1}{2}\frac{m_1 m_2}{m_1+m_2}v^2, \] where \(V\) is the velocity of their centre of mass, and \(v\) is their relative velocity. A shell of mass \((m_1+m_2)\) is fired with a velocity whose horizontal and vertical components are \(U, V\), and at the highest point in its path, the shell explodes into two fragments \(m_1, m_2\). The explosion produces an additional kinetic energy \(E\), and the fragments separate in a horizontal direction: shew that they strike the ground at a distance apart which is equal to \[ \frac{V}{g}\sqrt{2E\left(\frac{1}{m_1}+\frac{1}{m_2}\right)}. \]
State and prove the relation between the moment of inertia of a rigid body about any axis and its moment of inertia about a parallel axis through its centre of gravity. Find the moment of inertia in lb. ft\(^2\). units of a solid cylinder of length 10 inches and diameter 1 inch about an axis perpendicular to its length passing through its centre of gravity. The cylinder is composed of material weighing 0.27 lb. per cubic inch. Shew that if the cylinder is suspended in a horizontal position by two vertical strings of length 3 feet, one attached to each end, the time of a small oscillation in a horizontal plane is about 1.1 seconds.
The shape of the ground forming the bottom of a shallow tidal estuary is such that the area flooded is proportional to the square of the depth of the water at the entrance. The entrance channel has steep banks which may be taken as vertical. At low water the estuary and channel are just empty and as the tide rises the depth of water increases as a simple harmonic function of the time of which the periodic time is 12 hours. If the maximum height of the tide is 12 feet, the channel is 200 yards wide, and the area flooded at high tide 2 square miles, find the velocity of flow in the channel at half tide.