Question

Length of barrel of a gun is $l=5.0$ $m$ and mass of a shell fired is $m_o=45$ $kg$. During firing, combustion of the gun powder takes place at a constant rate $r=2.0\times {10}^3$ $kg/s$, the gun-powder is completely transformed into gas of average molar mass M$=5.0\times {10}^{-2}$kg/mol, temperature $T=1000$ $K$ of the gases remain fairly constant and displacement of the shell inside the barrel is proportional to $t^a$(t is time and a is a constant). Neglecting all forces other than force of the propellant gases during firing, find the muzzle velocity of the shell.

Answer 1

At time $dt$, mass of powder burned = $\frac{2.0\times {10}^3}{5.0\times {10}^{-2}}dt=0.4\times {10}^{5}moles\times dt$

Energy transmitted at $dt$ seconds = $\frac{3}{2}\times 0.4\times {10}^5\times 1000\times 8.314\left(\frac{3}{2}nRT\right)=4.98\times {10}^{8}J.dt$

Give,

$x=k{t}^a$ (k is constant)

$\frac{x}{dx}=\frac{t}{adt}$

$\frac{dx}{x}=\frac{adt}{t}$

$dt=\frac{t.dx}{ax}=\frac{4.98\times {10}^8\times t}{9\times 5}dx$

Total energy transfered = ${\int }_0^5$$\frac{4.98\times {10}^8\times t}{9\times 5}dx=$$\frac{4.98\times {10}^8\times t}{a}$

This energy = $\frac{1}{2}{m}_0{v}^2$

$\frac{1}{2}{m}_0{v}^2=$$\frac{4.98\times {10}^8\times t}{a}$

$v=\sqrt{\frac{2\times 4.98\times {10}^8\times t}{45\times a}}$

$v=0.47\times {10}^4\sqrt{\frac{t}{a}}$