A particle is projected freely under gravity: prove that its path is a parabola and that its velocity at any point is that due to a fall from rest at the level of the directrix. A stone is projected from a point \(P\) on the ground over a house so as just to clear the tops of the walls and the ridge of the roof; the breadth of the house is \(2a\), the height of each wall \(l\), and the height of the ridge \(h+l\). Find the position of \(P\) and the velocity of projection.
State the principle of the conservation of linear momentum. A particle of mass \(m\) lies on a smooth horizontal plane and is connected by smooth light inextensible strings, both taut, with particles of masses \(m'\) and \(m''\) also lying on the plane, the angle between the strings being \(2\alpha\). A blow is given to \(m\) in a direction bisecting the angle \(2\alpha\) so as to jerk the other masses into motion. Shew that the mass \(m\) begins to move in a direction \(\tan^{-1}\left(\frac{(m'-m'')\sin\alpha\cos\alpha}{m+(m'+m'')\sin^2\alpha}\right)\) with the bisector of the angle between the strings. Also find the kinetic energy of the system.
Two small heavy rings of masses \(m, m'\) are connected by a light rod, and slide upon a smooth vertical circular wire of radius \(a\), the rod subtending an angle \(\alpha\) at the centre; prove that the motion is the same as that of a simple pendulum of length \[ a(m+m')(m^2+2mm'\cos\alpha+m'^2)^{-1/2}, \] and find the pressures of the rings on the wire in any position if the system started from rest when the rod was vertical.
A particle moves in a straight line under the action of a force towards a fixed point in the line and varying as the distance from the point. Find the position of the particle at any instant, having given its initial position and velocity. A heavy particle of mass \(m\) is attached to the end of an elastic string of natural length \(a\) and modulus \(\lambda\), the other end of the string being fixed to a point \(A\). The particle is released from rest at \(A\) and falls under gravity: prove that the string will be extended during the interval of time \[ 2\{\pi - \tan^{-1}(2\lambda/mg)^{1/2}\}/(\lambda/ma)^{1/2}. \]
\(O\) is the circumcentre and \(P\) is the orthocentre of a triangle \(ABC\). Prove that the resultant of forces completely represented by \(AP, BP, CP\) is completely represented by \(2OP\).
A uniform isosceles triangle \(ABC\) rests with its plane vertical and its two equal sides \(AB, AC\) in contact with two smooth fixed pegs \(P\) and \(Q\). \(PQ\) is horizontal. Prove that the angle between \(BC\) and \(PQ\) is either zero or \[ \sin^{-1}\left[\frac{BC}{6PQ}(1+\cos A)\right]. \]
Two uniform rods \(AB, BC\) of equal weight but different lengths, are freely jointed together at \(B\) and placed in a vertical plane over two equally rough fixed pegs in the same horizontal line. The inclinations of the rods to the horizontal are \(\alpha, \beta\), and they are both on the point of slipping. Prove that the inclination \(\theta\) to the horizontal of the reaction at the hinge is given by \[ 2\tan\theta = \cot(\beta+\lambda) - \cot(\alpha-\lambda), \] where \(\lambda\) is the angle of friction at the pegs.
A mass of 160 lb. is attached to one end of a light rope, the other end of which is made fast at a point \(A\). The rope is elastic, obeying Hooke's law, and its breaking tension is 2000 lb. wt. If the rope does not break when the mass is dropped freely from \(A\), prove that the elongation of the rope under its breaking tension must exceed 19 per cent.
A tug leaves a port to intercept a liner, which is proceeding with uniform velocity \(u\) miles per hour on a straight course which, at the nearest point, is \(a\) miles from the port. The tug starts when the liner is \(b\) miles from the port and has not yet reached the nearest point. Prove that the least uniform speed the tug must have in order to reach the liner is \(\frac{au}{b}\). Prove also that if the tug can go \(v\) miles per hour \((u > v > \frac{au}{b})\), the liner is on a part of her course in which the tug can intercept her for \(\frac{2\sqrt{b^2v^2-a^2u^2}}{u^2-v^2}\) hours.
A rocket is fired vertically from the surface of the earth, and it may be assumed that when it has risen to a height of 50 miles the charge is all expended and air resistance has become negligible. Prove that at this altitude the rocket must have a velocity of nearly 7 miles per second if it is to escape from the attraction of the earth. Assume that the radius of the earth is 4000 miles, and that the attraction of the earth varies inversely as the square of the distance from its centre. Neglect the motion of the earth.