10273 problems found
A particle is projected with velocity \(V\) at an angle \(\alpha\) to the horizontal. Prove that its path is a parabola and find the length of its latus rectum. \par Two particles are projected simultaneously with velocities \(V, V'\) at angles \(\alpha, \alpha'\) with the horizontal from the same point \(O\) and in the same vertical plane. \(\alpha>\alpha'\) and \(V'\cos\alpha' > V\cos\alpha\). Prove that the particles arrive at the points of contact of the common tangent to the two trajectories at the same instant \(t\), where \[ t = \frac{VV'\sin(\alpha-\alpha')}{g(V'\cos\alpha'-V\cos\alpha)}. \]
Two particles \(A\) and \(B\) are in motion in a plane. Explain how to find the velocity of \(B\) relative to \(A\) in terms of the velocities of \(A\) and \(B\) referred to fixed axes in the plane. \par An aeroplane is flying at a uniform height with constant velocity \(v\). It is circling about a ship which is moving in a straight line with constant velocity \(ev\) where \(e<1\). Prove that the time taken to describe one such circle is \[ \int_0^{2\pi} \frac{a}{v(1-e^2)}\sqrt{1-e^2\sin^2\theta}\,d\theta, \] where \(a\) is the radius of the circle.
A particle is moving in a straight line so that \[ (2ksv^2+1)^3 = (3ktv^3+1)^2, \] where \(v\) is the velocity and \(s\) the distance described, both measured at the instant \(t\). \(k\) is a constant. Find the acceleration in terms of \(v\). \par If \(v=u\) when \(t=0\), shew that \(v=2u\) when \(s=\frac{3}{8ku^2}\).
When a body is immersed in liquid it is acted upon by an upward vertical force equal to the weight of liquid displaced and passing through the centroid of the volume immersed. Consider the case of a solid circular cylinder of radius \(a\) partly immersed in liquid in a hollow circular cylinder of radius \(b\). Assuming that the solid cylinder does not come in contact with the base of the hollow cylinder and that it is always partly immersed, shew that the motion of the solid cylinder is simple harmonic with period \(2\pi\sqrt{\frac{l(b^2-a^2)}{gb^2}}\), where \(l\) is the length of the solid cylinder immersed when floating in equilibrium. It may be assumed that the axes of both cylinders are vertical.
A particle is suspended from a fixed point by a light inextensible string of length \(l\). If the particle receives a horizontal velocity \(u\), find conditions such that the string shall become slack in the subsequent motion, and prove that in this case the string is slack for a time \(\sqrt{\frac{8l}{g}}\sin\phi \sin 2\phi\) where \(3\cos\phi = \frac{u^2}{gl}-2\).
A uniform straight rod at rest receives simultaneously an impulse \(P\) in the direction of its length and an impulse \(Q\) at one end in a direction perpendicular to the rod. If the initial velocities of the two ends are in perpendicular directions, determine the relation between \(P\) and \(Q\).
State and prove a rule for expressing \[ \frac{P(x)}{Q(x)} \] as the sum of a polynomial and partial fractions, where \(P\) and \(Q\) are polynomials, and \(Q\) has no repeated factors. \par Express in this form \[ \frac{(x-a)(x-b)(x-c)(x-d)}{(x+a)(x+b)(x+c)(x+d)}, \] (i) when \(a,b,c,d\) are all unequal, (ii) when they are all equal.
Shew that if \(x\) is added to all the elements of any determinant, the resulting determinant has the value \(A+Bx\), where \(A\) and \(B\) are independent of \(x\). \par Shew that for the determinant \[ \begin{vmatrix} a & b & b & b \\ p & a & b & b \\ q & r & a & b \\ s & t & u & a \end{vmatrix} \] \(A = \Delta\) and \(B=\Delta-(a-b)^4\), where \(\Delta\) is the value of the determinant; and hence evaluate \[ \begin{vmatrix} a & b & b & b \\ c & a & b & b \\ c & c & a & b \\ c & c & c & a \end{vmatrix}. \]
Find the condition that the \(n\)th term in the expansion of \((1-x)^{-k}\) exceed the next, assuming that \(k > 0\) and \(0 < x < 1\). Hence find the position and value of the greatest terms of the expansions of
Solution: The ratio of the \((n+1)\)th term in the expansion of \((1-x)^{-k}\) to the \(n\)th term is \(\displaystyle \frac{(-k-n)}{n+1}(-1)x = \frac{n+k}{n+1}x\), which is increasing if \((n+k)x > n+1 \Leftrightarrow kx-1 > (1-x)n \Leftrightarrow \frac{kx-1}{1-x} > n\). Therefore the largest element is when \(\displaystyle \left \lfloor \frac{kx-1}{1-x} \right \rfloor\).
The function \(\cot\theta + k\sec\theta\), (\(k>0\)), has a turning value when \(\theta=\alpha\). Find a cubic satisfied by \(\sin\alpha\), and shew, by a graph or otherwise, that just one root of this cubic gives real values of \(\alpha\). \par Shew that of the two turning values of the function between \(0\) and \(\pi\) one is a minimum and the other a maximum, and sketch the graph of \(\cot\theta+2\sec\theta\) for the range \(-\pi < \theta < \pi\).