Air is to be compressed into a chamber of volume \(V\) by means of a pump. The pump has a cylinder of volume \(v\); it takes in air from the atmosphere, compresses it adiabatically, and forces it into the chamber through a non-return valve. Show that, neglecting the clearance of the pump, the pressure in the chamber after \(n\) strokes of the pump is \(\left(1 + n\frac{v}{V}\right)^\gamma\) times the pressure of the atmosphere. (\(\gamma\) is the ratio of specific heat at constant pressure to specific heat at constant volume.)
Prove that, if \(r < a-b\), there are eight normals to the ellipse \(x^2/a^2 + y^2/b^2=1\) which are tangents to the circle \(x^2+y^2=r^2\); and shew that the corresponding tangents to the ellipse touch one or other of the circles \(x^2+y^2=R^2\), where \[2R = \{(a+b)^2 - r^2\}^{\frac{1}{2}} \pm \{(a-b)^2-r^2\}^{\frac{1}{2}}.\]
A particle is projected from \(O\) in the direction \(OT\) with velocity \(V\), and at the same instant an equal particle is let fall from \(T\). Shew that they collide, and that if they coalesce after collision, the path described is that of a particle projected at the same instant from the middle point of \(OT\) in the same direction \(OT\) with velocity \(\frac{1}{2}V\).
Prove that the sum of the squares of the medians of a triangle \(ABC\) is \(\frac{3}{4}(a^2+b^2+c^2)\). If \(B=55^\circ\), \(C=23^\circ 30'\) and the median \(AD = 400\) feet, find to the nearest foot the lengths of the median \(BE\).
Prove that kinetic energy is always destroyed in the impact of inelastic particles. A mass \(M\) is attached to one end of a fine inextensible string, and particles each of mass \(m\) are tied to the string at regular intervals \(a\). The string is placed over a fixed smooth pulley and the particles coiled together on a table. Shew that if the mass falls freely through a distance \(a(M+m)(4M-m)/6Mm\) before the string becomes taut, the \((r+1)\)th particle is dragged off the table with velocity \[ \left\{ga \frac{M+mr}{M+m(r+1)} \frac{4M - m(2r+1)}{m}\right\}^{\frac{1}{2}}. \]
The following table gives the volume (\(v\)) of one pound of dry saturated steam at different pressures (\(p\)):
Prove that the two circles \(3x^2+3y^2+6ax = a^2\) and the hyperbola \(6x^2 - 3y^2 = 2a^2\) are so related that any pair are polar reciprocals with regard to the third.
A train starts from a station \(A\) with an acceleration 1 foot per second per second, the acceleration decreasing uniformly for two minutes, at the end of which time the train has acquired its full velocity: the full velocity is maintained for 5 minutes when the brakes are applied producing a constant retardation of 3 feet per second per second, bringing the train to rest at the station \(B\). Draw the acceleration-time curve and deduce the velocity-time curve. Find the maximum velocity attained and the distance between the stations \(A\) and \(B\).
The bisectors of the angles of the triangle \(ABC\) cut the opposite sides in \(D, E, F\). Find the lengths of the segments of the sides, and prove that the ratio of the area of the triangle \(DEF\) to that of the triangle \(ABC\) is \[ \frac{2abc}{(b+c)(c+a)(a+b)}. \]
A particle moves in a plane curve; determine the tangential and normal components of its acceleration. A particle moves under gravity on a cycloid, whose axis is vertical and vertex downwards, starting from rest at the cusp. Shew that its acceleration is \(g\), directed towards a point which moves in a horizontal line with constant velocity \(\sqrt{ga}\), where \(a\) is the radius of the generating circle of the cycloid. Shew also that the hodograph is a circle through the pole of the hodograph, described with uniform speed; and connect the two results obtained.