From the vertex of the parabola \(y^2-4ax=0\), lines are drawn parallel to the tangents to the curve, prove that the locus of the points where they meet the corresponding normals is the curve \(y^4(x^2+y^2) = ax(x^2+2y^2)\).
State and prove the acceleration property of the hodograph. Determine the hodographs of (1) a projectile describing a parabolic path, (2) a particle on the inside of the rim of a cartwheel which is travelling with uniform speed along a straight road. The thickness of the rim is to be taken as \(\frac{1}{10}\) of its external radius. In each case draw also the actual paths of the bodies, and indicate corresponding points on path and hodograph. A radial force of \(1\frac{1}{2}\) ounces weight is needed to detach the particle, weighing 2 oz., from the rim. If the cart is travelling at 4 miles an hour, and the particle is in limiting equilibrium at the top of its path, find the radius of the wheel.
A circular iron plate conducts 15 Pound-Centigrade thermal units per minute through its thickness per square inch of surface. If the thermal conductivity of iron be 0.00084 in inch units and the coefficient of expansion be 0.0000122 per degree Centigrade, find the radius of curvature of the heated plate, if it be flat when cold.
Shew that \[ \frac{2n+(n+1)x}{2n+(n-1)x} < \sqrt[n]{(1+x)}, \] if \(x > 0\) and \(n > 1\). Shew also that the difference between the two functions, when \(x\) is small, is approximately \[ \frac{n^2-1}{12n^3}x^3. \]
The effective horse-power required to drive a ship of 15,000 tons at a steady speed of 20 knots is 25,000. Assuming that the resistance consists of two parts, one constant and one proportional to the square of the speed, these parts being equal at 20 knots, and that the propeller thrust is the same at all speeds, find the initial acceleration when starting from rest, and the acceleration when a speed of 10 knots is attained. Shew that this speed is attained from rest in about 93 seconds and the distance traversed is about 271 yards. [One knot = 100 ft. per minute; \(\log_e 4 = 1.3863\), \(\log_e 3 = 1.0986\).]
From the focus \(S\) of an ellipse whose eccentricity is \(e\), radii \(SP, SQ\) are drawn at right angles to one another and the tangents at \(P\) and \(Q\) meet at \(T\). Shew that the locus of \(T\) is a hyperbola, parabola or ellipse according as \(e >=< \frac{1}{\sqrt{2}}\).
Two bar magnets are each of length 50 cm., but their pole strengths are 100 and 50 units respectively. If they are placed so that their poles form a regular tetrahedron, find the magnetic force at the centre of gravity of the tetrahedron.
Prove that, if \(f\) is a homogeneous polynomial in \(x\) and \(y\), of degree \(n\), then
A naval target is rising and falling on the waves with simple harmonic motion, the height of the waves from crest to trough being 6 feet and their period 10 seconds. At a range of 3000 yards a gun is sighted correctly and fired whilst the target is at the crest of a wave: by what distance will the shot miss the centre of the target, if the horizontal velocity of the shot be 2000 ft. per sec.?
If \(z=(1-2ax+a^2)^{-\frac{1}{2}}\), prove that \[ \frac{\partial}{\partial x}\left\{(1-x^2)\frac{\partial z}{\partial x}\right\} + \frac{\partial}{\partial a}\left\{a^2 \frac{\partial z}{\partial a}\right\}=0. \]