A body is projected from the ground with velocity \(u\) at inclination \(\alpha\) to the horizontal. At the highest point of the trajectory the body is broken into two parts by an internal explosion which creates E foot-pounds of energy without altering the direction of motion. Shew that the distance between the parts when they reach the ground is \[ 2\left(\frac{E}{mg}\right)^{\frac{1}{2}} u \sin\alpha, \] where \(m\) is the harmonic mean of the masses of the parts.
A sphere of mass \(m\) impinges directly on a sphere of mass \(m'\) at rest on a smooth table. The second sphere then strikes a vertical cushion at right angles to its path. Shew that there will be no further impact of the spheres if \(m(1+e'+ee') < em'\); where \(e, e'\) are the coefficients of restitution between the spheres and between the sphere and the cushion.
Shew that the roots of \[ (x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0 \] are real, and cannot be equal unless \(a=b=c\). Eliminate \(x,y\) from \[ x(x-y)=a^2, \quad y(x+y)=b^2, \quad 1/x^2+1/y^2=1/c^2. \]
Find the conditions that \(ax^2+2bx+c\) should be positive for all real values of \(x\). Find the greatest value of \[ (a-x)(x+\sqrt{b^2+x^2}). \]
If \[ (1+x)^n = c_0+c_1x+c_2x^2+\dots \] find \[ c_0 - c_1 + c_2 - \dots + (-1)^n c_n. \] Prove that the coefficient of \(x^n\) in the expansion in powers of \(x\) of \[ \frac{1}{(1-x)(1-x^2)(1-x^5)} \] is \((n+1)^2\). (Note: The original text had a typo in the denominator, \(x^5\) vs \(x^3\). Transcribed as written. Also, the claim regarding the coefficient seems incorrect for the given function.)
Prove Fermat's theorem that \(a^n-x\) is divisible by \(n\) if \(n\) is a prime and \(x\) any positive integer. Prove that \(x^{45}-x\) is divisible by 69.
Define the differential coefficient of a function of \(x\). If \(f(x)\) is positive shew that \(f(x)\) is increasing. Prove that \(x\log x > x-1\) if \(x\) is positive. Differentiate \(x^x, \tan^{-1}\left(\frac{x\sin\alpha}{1-x\cos\alpha}\right)\).
If \(x=r\cos\theta, y=r\sin\theta\), find the values of \(\frac{\partial r}{\partial x}\) and \(\frac{\partial r}{\partial y}\). Transform the variables from \(x, y\) to \(r, \theta\) in \[ x\frac{\partial u}{\partial y} - y\frac{\partial u}{\partial x}. \]
Prove that if \(p\) is the perpendicular from the origin on the tangent to a curve \(r=f(\theta)\), \[ \frac{1}{p^2} = \frac{1}{r^2} + \frac{1}{r^4}\left(\frac{dr}{d\theta}\right)^2. \] Prove that the feet of the perpendiculars from the origin on the normals to the curve \(r^2=a^2\cos 2\theta\) lie on the curve \[ 4r^2/a^2 = \cos 2\theta + \cos(\frac{1}{3}\pi+\theta). \] (Note: The constant in the cosine term is transcribed as it appears.)
Trace the curve \(x^4 - x^2y+y^3=0\).