An engine moves at a steady velocity \(v\) along level ground when working at a constant horse-power \(H\). When moving up a plane inclined at a small angle to the horizontal its steady velocity under the same horse-power is \(v'\). If the engine starts down the same incline with velocity \(v'\) and moves for \(t\) seconds with a constant acceleration until it reaches its steady velocity down the plane corresponding to the same horse-power, \(H\), shew that the distance travelled in these \(t\) seconds is \(v't(2v-v')/v\). Assume that the frictional resistance is constant throughout.
A ball is dropped from the top of a tower 100 feet high. At the same moment a ball of equal mass is thrown from a point on the ground 50 feet from the foot of the tower so as to strike the first ball when just half-way down. Find the initial velocity of projection of the second ball and the direction of projection. If the two balls coalesce how long will they take to reach the ground?
A number of particles lie on the equiangular spiral \(r=Ae^{\theta \tan\alpha}\) and are in motion. Prove that, if the particles continue to lie on an equiangular spiral, \(\mu\) (the component of velocity of a particle normal to the curve) is of the form \[\mu = r(p+q\log r),\] where \(p, q\) are functions of \(t\) only. If \(p, q\) are both constant, find relations connecting \(p, q, A\) and \(\alpha\) with \(t\).
A particle of mass \(m\) is moving in the axis of \(x\) under a central force \(\mu mx\) to the origin. When \(t=2\) seconds, it passes through the origin, and when \(t=4\) seconds, its velocity is 4 feet per second. Determine the motion and shew that, if the complete period is 16 seconds, the semi-amplitude of the path is \(\frac{32\sqrt{2}}{\pi}\) feet.
A sledge hammer consists of an iron rectangular block 6 ins. \(\times\) 2 ins. \(\times\) 2 ins. A central circular hole of 1 inch diameter is bored through it at right angles to one of its longer faces and a light shaft 3 ft. long of wood is fitted into it. Find the moment of inertia of the hammer about a line drawn through the mid-point of the far end of the shaft normal to the axis of the shaft and parallel to the small face of the block. (Take the density of iron as 7.0.)
In any triangle, prove that the centre of the nine-points circle bisects the straight line joining the orthocentre to the centre of the circumscribing circle. If an ellipse is inscribed in the triangle with one focus at the orthocentre, prove that the other focus is at the centre of the circumscribing circle, and that the major axis is equal to the radius of this circle.
A, Q, B, P, C are five points in a straight line such that A, P are harmonically conjugate with respect to B, C; and also C, Q with respect to A, B. Prove that \[ 4AC.QP=3AP.QC. \]
S is the focus of a parabola, and the normal at P meets the axis in G. Prove that \(SG=SP\). F and H are points on the axis such that PG bisects the angle FPH. Prove that \(SF.SH=SG^2\).
A and B are given points. A central conic of given eccentricity is drawn touching AB, and such that its directrix passes through A and the corresponding latus rectum passes through B. For all such conics, prove that the locus of the corresponding focus is a circle.
Show that in a given direction two straight lines can be drawn touching both of two given spheres, provided that the lines through the centres in this direction are at a distance apart intermediate between the sum and the difference of the radii.