Question

A rocket with an initial mass of $M=2000\text{kg}$ is launched vertically upwards from rest, under gravity. The rocket burns fuel at the rate of $20\text{kg/s}$. The burnt matter is ejected vertically downwards with a speed of $4000\text{m/s}$ relative to the rocket. Velocity of the rocket after $60$ seconds is $2000\text{m/s}$. Find the mass of rocket gas ejected in $60$ seconds. [Take $g=10{\text{m/s}}^2$]

• A. $420.62\text{kg}$
• B. $955.86\text{kg}$
• C. $590.62\text{kg}$
• D. $500.92\text{kg}$

Answer 1

The correct option is B $955.86\text{kg}$

Given,
Mass of rocket initially, $M=2000\text{kg}$
Fuel burn rate $\mu =20\text{kg/s}$
Burning matter ejected with a speed of $v_r=4000\text{m/s}$
Initial velocity of rocket $u=0$
Velocity of rocket after $60$ seconds $V=2000\text{m/s}$

Mass of rocket after $60\text{sec}$,
$m=M-\mu t=2000-\mu \times 60$
Using the velocity equation, $V=u-gt+v_r\ln \left(\frac{M}{m}\right)$
$2000=0-10\times 60+4000\ln \left(\frac{2000}{2000-60\mu }\right)$
or $\mu =15.931\text{kg/s}$
Therefore, mass ejected in $60$ seconds $=\mu \times 60$
$=955.86\text{kg}$