Question

In a gravity free space, a spacecraft of mass $10\text{kg}$, having $20\text{kg}$ of fuel is initially at rest. The fuel is ignited and the exhaust velocity is $0.1\text{km/s}$. Then, speed gained by the spacecraft when the fuel is fully consumed is

• A. $109\text{km/s}$
• B. $10.9\text{km/s}$
• C. $109\text{m/s}$
• D. $1.09\text{km/s}$

Answer 1

The correct option is C $109\text{m/s}$

Given,
Mass of spacecraft without fuel $=10\text{kg}$
Mass of fuel in the spacecraft $=20\text{kg}$
Exhaust velocity of the fuel $v_r=0.1\text{km/s}$

Speed gained by the spacecraft is given by
$v=u-gt+v_r\ln \left(\frac{m_0}{m}\right)$
Here, initial velocity $u=0$
$g=0$ (because gravity is absent)
$m_0=$ Initial total mass of the space craft $=30\text{kg}$
and when the whole fuel is exhausted, mass of the space craft $\left(m\right)=10\text{kg}$

$\Rightarrow v=0.1\times {10}^3\ln \left(\frac{30}{10}\right)\approx 109\text{m/s}$