Question

A rocket with an initial mass of $1000\text{kg}$ is launched vertically upwards from rest under gravity. The rocket burns fuel at the rate of $10\text{kg}$ per second. The burnt matter is ejected vertically downwards with a speed of $2000\text{m/s}$ relative to the rocket. If burning ceases after one minute, the maximum velocity of the rocket is

• A. $4000.6\text{m/s}$
• B. $2500.4\text{m/s}$
• C. $4500.2\text{m/s}$
• D. $1232.6\text{m/s}$

Answer 1

The correct option is D $1232.6\text{m/s}$

Given, initial mass of rocket, $m_0=1000\text{kg}$
Initial speed $u=0$
Rate of burning of fuel $=10\text{kg/s}$
Speed of burnt matter with respect to rocket $v_r=2000\text{m/s}$
time of burning of fuel, $t=1\text{minute}=60\text{s}$


Final mass of rocket after $60\text{sec}$
$m=1000-10\times 60$
$m=400\text{kg}$

We know that,
$v=u-gt+v_r\ln \left(\frac{m_0}{m}\right)$
$\Rightarrow v=0-10\times 60+2000\ln \left(\frac{1000}{400}\right)$
$\Rightarrow v=-600+1832.6$
$\Rightarrow v=1232.6\text{m/s}$
The maximum velocity of rocket is $1232.6\text{m/s}$