The resistance of the air to bullets of given shape varies as the square of the velocity and the square of the diameter, and for a particular bullet (diameter 0.3") is 40 times the weight at 2000 f.s. For an exactly similar bullet of the same material (diameter 0.5") show that the velocity will drop from 2000 f.s. to 1500 f.s. in about 500 yards, assuming the trajectory horizontal. [\(\log_e 10 = 2.30\).]
A motor car of weight \(W\) is being decelerated at rate \(f\) by application of the brakes. Determine the reactions between the wheels and the road, when the masses of the wheels may be neglected, and the brakes are applied only to the rear wheels. The centre of gravity of the car is at a height \(h\) above the road, and at horizontal distances \(a\) from the rear and \(a'\) from the front wheels. Shew that the maximum deceleration, which can be obtained without skidding any wheel, is greater by the factor \(\dfrac{a+a'+h\mu}{a'}\) for a car braked on all four wheels than for a car braked on the rear wheels only, where \(\mu\) is the coefficient of friction between the tyres and the road. If for example \(a'=a=2h, \mu=0.9\), four-wheel braking has the advantage in a factor of \(2.45\). Explain in general terms how it comes about that this factor can be greater than 2, when \(a=a'\).
The foci of an ellipse are the points \((0,0)\), \((c,0)\), and the ellipse passes through the point \((\frac{7}{4}c, \frac{3}{4}c)\). Find its eccentricity.
Prove that, if the equation \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \] is satisfied by \(u=f(x,y)\), a function homogeneous in \(x, y\) of degree \(n\), it is also satisfied by \[ u = \frac{f(x,y)}{r^{2n}}, \] where \(r^2=x^2+y^2\).
A shot is fired with initial velocity \(V\) at a mark in the same horizontal plane; show that if a small error \(\epsilon\) is made in the angle of elevation, and an error \(2\epsilon\) in azimuth, the shot will strike the ground at a distance from the mark \(\dfrac{\pi V^2 \epsilon}{90g}\). Show also that if the angle of elevation is less than about \(31\frac{1}{2}^\circ\) an error in elevation will cause the shot to miss the mark by a greater amount than an equal error in azimuth.
Define the hodograph, establish its principal properties and its importance in practical applications. Prove in particular by its means that a particle describing any curve with a (variable) velocity \(v\) has at any moment an acceleration \(v^2/\rho\) along the normal to the curve. Shew further either that a particle describing an ellipse under a force to the centre must be attracted by a force varying directly as the distance from the centre, or that a particle describing an ellipse under a force to a focus must be attracted by a force varying inversely as the square of the distance from the focus.
Differentiate \(\sin^{-1} \{2x \sqrt{(1-x^2)}\}\), \(a^{x \log a}\). If \(x\) is large, show that the differential coefficient of \((1+\frac{1}{x})^x\) is approximately \(e/2x^2\).
Find the shortest distance between the point \((b,0)\) and points of the parabola \(y^2 = 4ax\), distinguishing the cases \(b>2a\), \(b<2a\).
Two equal spheres of mass \(9m\) are at rest and another sphere of mass \(m\) is moving along their line of centres between them. How many collisions will there be if the spheres are perfectly elastic?
Sketch the curve \[ y^2 = a^2 (2x^2-a^2)/4x^2. \] Find the equation of the tangent at the point \((\frac{1}{2}a, \frac{1}{2}a)\), and show that it touches the curve at a second point.