State Newton's Laws of Motion and shew how they give rise to the equation \(P=mf\) and to the absolute and gravitational units of force. A particle of variable mass is in motion in a straight line under the action of a force whose magnitude in absolute units at any instant is twice the product of the mass of the particle and its acceleration at that instant. Prove that the impulse of the force in any interval is proportional to the change in the square of the velocity in that interval. It may be assumed that the particle does not come to rest at any instant in the interval considered.
A lift moves vertically upwards from rest with uniform acceleration \(f( < g)\) and as it starts to move a ball is dropped on to the floor of the lift. If the lift overtakes the ball when the latter is at the highest point (in space) of its first bounce, shew that the coefficient of restitution between the ball and the floor of the lift is \(\frac{f}{g-f}\) and that the time that elapses between the instant of starting and the instant of overtaking is \(\frac{\sqrt{2h(g+f)}}{g-f}\) where \(h\) is the initial height of the ball above the floor of the lift. (Note: The scanned document's formula for the coefficient of restitution is unclear, this transcription uses the physically derived correct value.)
The time taken by a shell of mass \(m\) fired with speed \(V\) at an angle \(\alpha\) to the horizontal to reach the highest point of its trajectory is \(t\) seconds. \(\frac{3t}{2}\) seconds after firing, the shell is split into two parts of equal mass by an explosion which increases the energy of the subsequent motion by \(\frac{mV^2}{8}\). Immediately after the explosion it is observed that the horizontal velocity of one part has been increased and its vertical velocity annulled and that both parts continue to move in the same vertical plane as that in which the motion was taking place before the explosion. Prove that the two parts strike the horizontal plane through the point of projection at a distance apart \[ \frac{V^2\sin 2\alpha}{8g}(3\sqrt{3}+2-\sqrt{7}). \]
Prove that, if \(p/q\) is a fraction in its lowest terms, then integers \(r\) and \(s\) can be found such that \(qr-ps=1\). Prove that, if \(p/q\) and \(r/s\) are fractions such that \(qr-ps=1\), then the denominator of any fraction whose value lies between \(p/q\) and \(r/s\) is at least \(q+s\).
Prove that \[ a^2+b^2+c^2-bc-ca-ab = (a+\omega b+\omega^2 c)(a+\omega^2 b + \omega c), \] where \(\omega\) is a complex cube root of 1. Prove that, if \[ (b-c)^n+(c-a)^n+(a-b)^n \] is divisible by \(\Sigma a^2 - \Sigma bc\), then \(n\) is an integer not a multiple of 3. Prove that, if the same expression is divisible by \((\Sigma a^2 - \Sigma bc)^2\), then \(n\) is greater by one than a multiple of 3.
Prove that, if \((1+x)^n = c_0+c_1x+\dots+c_nx^n\), then \[ c_0c_2+c_1c_3+\dots+c_{n-2}c_n = \frac{(2n)!}{(n-2)!(n+2)!}, \] \[ \frac{c_0}{1} - \frac{c_1}{2} + \frac{c_2}{3} - \dots + (-1)^n\frac{c_n}{n+1} = \frac{1}{n+1}, \] \[ \frac{c_0}{1^2} - \frac{c_1}{2^2} + \frac{c_2}{3^2} - \dots + (-1)^n\frac{c_n}{(n+1)^2} = \frac{1}{n+1}\left(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n+1}\right). \]
Find an equation connecting the expressions \[ \cos A + \cos B + \cos C, \] \[ \sin A \sin B \sin C, \] where \(A, B\) and \(C\) are the angles of a triangle. Prove that the sum of the cosines of the angles of a triangle is greater than 1 and not greater than \(\frac{3}{2}\).
Prove that, if \(H\) and \(O\) are the orthocentre and circumcentre of a triangle \(ABC\), \[ OH^2=R^2(1-8\cos A \cos B \cos C), \] where \(R\) is the radius of the circumcircle. Prove that if \(K\) is the middle point of \(OH\), \[ AK^2+BK^2+CK^2 = 3R^2-\frac{1}{4}OH^2. \]
Prove that the function \[ \frac{\sin^2 x}{\sin(x-\alpha)}, \] where \(0 < \alpha < \pi\), has infinitely many maxima equal to 0 and minima equal to \(\sin\alpha\). Sketch the graph of the function.
If \(x, y, z\) are connected by an equation \(\phi(x,y,z)=0\), explain the meaning of the partial differential coefficient \(\partial z/\partial x\), and express it in terms of the partial differential coefficients of the function \(\phi(x,y,z)\). If \[ \frac{x^2}{a^2+z} + \frac{y^2}{b^2+z} = 1, \] where \(a\) and \(b\) are constants, prove that \[ \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2 = 2\left(x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y}\right). \]