Question

Scientists of $ISRO$ want to achieve a velocity of rocket $100\text{m/s}$ after time $t=5\text{sec}$. Velocity of ejected burnt fuel is $20\text{m/s}$ w.r.t the rocket. What should be the rate of consumption of fuel (approximately) so that the rocket will achieve the desired velocity?
Initial mass of rocket is $2000\text{kg}$. [Take $g=10{\text{m/s}}^2$]

• A. $300\text{kg/s}$
• B. $400\text{kg/s}$
• C. $100\text{kg/s}$
• D. $200\text{kg/s}$

Answer 1

The correct option is B $400\text{kg/s}$

Given,
Initial mass of rocket $m_0=2000\text{kg}.$
Velocity of rocket after $5\text{sec}$ $v=100\text{m/s}$
Initial velocity $u=0\text{m/s}$
Velocity of ejected burnt fuel $v_r=20\text{m/s}$
As we know, $v=u-gt+v_r\ln \left(\frac{m_o}{m}\right)$
Here $m=m_0-\mu t$
$=2000-\mu \times 5$ ...(1)
Then,
$100=0-10\times 5+20\times \ln \left(\frac{2000}{2000-5\mu }\right)$
$\Rightarrow 7.5=\ln \left(\frac{2000}{2000-5\mu }\right)$
$\Rightarrow 1808.04=\frac{2000}{2000-5\mu }$
$\Rightarrow \mu =399.78\text{kg/s}\approx 400\text{kg/s}$