A bullet of mass \(\frac{1}{2}\) oz., moving in the path \(BC\), strikes and embeds itself in \(M\), a mass of 5 lbs. which is suspended from \(A\) by a light, flexible and inextensible string. The two masses then move together, as shewn, with common velocity \(V=5\) f.p.s. What is the mean value of the extra tension imposed on the string, if the bullet becomes fixed in the mass \(M\) within \(\frac{1}{2000}\) sec. after impact?
Describe the graphical methods employed in dynamics for the determination of any two of the quantities velocity, acceleration, distance, time from a knowledge of the relation between the other two, paying particular attention to scales of measurement.
\(PCP'\) is a diameter of the ellipse \(\displaystyle\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\); \(CD\) is a conjugate semi-diameter. Prove that the locus of the centre of the circle \(PDP'\) is \[ 4(a^2x^2 + b^2y^2)^3 = (a^2-b^2)^2(a^2x^2-b^2y^2)^2. \]
Prove that if \(f(x)\) and its first two derivatives are continuous in \(0 \le x \le a\) (\(a>0\)), and \(x, x+h\) are any two points of this interval, then \[ f(x+h) = f(x) + hf'(x) + \frac{1}{2}h^2f''(x+\theta h), \] where \(\theta\) is some number satisfying \(0 < \theta < 1\). By taking \(h=-x\), or otherwise, prove that, if \(f(0)=0\), and \(f''(x)>0\) in \(0 < x < a\), then \(f(x)/x\) is an increasing function of \(x\) in this interval. Explain the geometrical significance of this result. Deduce that \((\sin x)/x\) decreases as \(x\) increases from \(0\) to \(\pi\).
An aeroplane flies horizontally at 90 m.p.h. through rain which is falling vertically at a rate of \(2\frac{1}{2}\) lbs. per sq. foot of ground surface per hour. Assuming that each raindrop, on striking the aeroplane, acquires its velocity through the air, and that the aeroplane is exposing a surface equivalent to 200 sq. ft. of horizontal surface, calculate the extra resistance which the aeroplane experiences from this cause.
Prove the principle of conservation of linear momentum and develop the method of determining the motion of smooth spheres after impact, stating Newton's rule. Shew that in the case of two spheres an equivalent hypothesis is that the impulse along the line of centres during restitution is \(e\) times the impulse during compression, when \(e\) is the coefficient of restitution. Use the latter hypothesis to solve the following problem. Three equal similar spheres of mass \(m'\) are suspended by equal vertical threads so that their centres are at the corners of an equilateral triangle in a horizontal plane. A fourth smooth sphere, of mass \(m\), falls vertically so as to strike the other spheres simultaneously. Determine the velocities immediately after impact, having given the velocity \(u\) of the fourth sphere and the angle \(\theta\) which the lines of centres make with the vertical at the instant of striking.
Find the stationary values of \[ y = 10 \frac{x^2+3x}{2x^2+13x-7}. \] Give a rough sketch of the graph of this equation.
Define the partial derivatives \(f_x, f_y, f_{xx}, f_{xy}, f_{yx}, f_{yy}\) of a function \(f(x,y)\). \(f(x,y)\) is defined to be \(\displaystyle\frac{x^2y-xy^2}{x^2+y^2}\) when \(x\) and \(y\) are not both zero, and 0 when \(x=y=0\). Calculate from the definition the values of the six partial derivatives at the point \(x=y=0\). Verify that \(f_{xy}(x,y)=f_{yx}(x,y)\) when \(x\) and \(y\) are not both zero, but that \(f_{xy}(0,0) \ne f_{yx}(0,0)\).
Shew that all points in a vertical plane, which can be reached by shots fired with velocity \(v\) from a fixed point at a distance \(c\) from the plane, lie within a parabola of latus rectum \(\frac{2v^2}{g}\), whose focus is at a distance \(\frac{c^2g}{2v^2}\) vertically below the foot of the perpendicular on the plane from the point of projection.
A smooth wedge of mass \(M\) resting on a horizontal plane is subject to smooth constraints so that it can only move along the plane in a direction at right angles to the intersections of its slant faces with the plane. A particle of mass \(m\) is moving along a face of the wedge which is inclined to the horizontal at an angle \(\alpha\) so that the component velocity of the particle perpendicular to the line of greatest slope is \(u\). Shew that the wedge moves with constant acceleration and that the path of the particle on the surface of the wedge is a parabola of latus rectum \[ \frac{2u^2}{g} \frac{M+m\sin^2\alpha}{(M+m)\sin\alpha}. \] Verify that the principle of conservation of energy holds good in this case. Solve the same problem when the wedge, instead of being free to move, is made to move in the same direction as before with constant velocity \(V\).