Question

A graph of mass ejected vs time is shown in the figure below. [Given $\ln \left(\frac{50}{49}\right)=0.02$]


Find the velocity of the rocket after $5\text{s}$, if initial launch velocity is $200\text{m/s}$ and velocity of ejected fuel w.r.t the rocket is $20\text{m/s}$. [Take $g=10{\text{m/s}}^2$ and initial mass of the rocket is $500\text{kg}$ ].

• A. $20.4\text{m/s}$
• B. $150.4\text{m/s}$
• C. $30.4\text{m/s}$
• D. $15.6\text{m/s}$

Answer 1

The correct option is B $150.4\text{m/s}$

Given, $V_o=$ initial launch velocity = $200\text{m/s}$
$V_r=$ velocity of ejected fuel w.r.t rocket $=20\text{m/s}$
$m_0=$ initial mass $=200\text{kg}$
From the graph, rate of mass ejected
$\frac{dM}{dt}=\mu =\frac{10-0}{5-0}=2\text{kg/s}$

As we know, velocity after time $t$
$V=V_o-gt+V_r\ln \left(\frac{m_o}{m}\right)$
Here, $V=200-10\times 5+20\times \ln \left[\frac{500}{500-2\times 5}\right]$
$=150+20\times \ln \left[\frac{50}{49}\right]$
$=150+20\times 0.02$
$\therefore V=150.4\text{m/s}$