Question

$PSLV$ decides to send a rocket to the moon for which it requires a velocity of $1232.6\text{m/s}$ after $1\text{min}$. The rocket burns fuel at the rate of $10\text{kg/s}$. The burnt matter is ejected vertically downwards with a speed of $2000\text{m/s}$ relative to the rocket. Calculate the initial mass of rocket. [Take $g=10{\text{m/s}}^2$]

• A. $400\text{kg}$
• B. $1000\text{kg}$
• C. $500\text{kg}$
• D. $200\text{kg}$

Answer 1

The correct option is B $1000\text{kg}$

Given,
Velocity of rocket after $1\text{min}$, $v=1232.6\text{m/s}$
Fuel burn rate =$\frac{dM}{dt}=10\text{kg/s}=\mu$
Speed of ejecting burnt matter $V_r=2000\text{m/s}$
Initial mass $=m_0$

Using the velocity equation: $v=u-gt+v_r\ln \left(\frac{m_0}{m}\right)$
Here,
$m=m_0-\mu t=m_0-10\times 60=m_0-600$
Therefore,
$1232.6=0-10\times 60+2000\times \ln \left(\frac{m_0}{m_0-600}\right)$
$\Rightarrow 0.9163=\ln \left(\frac{m_0}{m_0-600}\right)$
$\Rightarrow m_0=1000\text{kg}$