(C) The velocity of a rocket at any instant is given by the Tsiolkovsky rocket equation: $v = v_e \ln \left( \frac{m_0}{m} \right)$.Here,$v_e$ is the

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(C) The velocity of a rocket at any instant is given by the Tsiolkovsky rocket equation: $v = v_e \ln \left( \frac{m_0}{m} \right)$.Here,$v_e$ is the exhaust velocity relative to the rocket,$m_0$ is the initial mass,and $m$ is the final mass.Given that the final mass $m = \frac{m_0}{2}$,we substitute this into the equation:$v = v_e \ln \left( \frac{m_0}{m_0/2} \right)$$v = v_e \ln(2)$Since $\ln(2) \approx 0.693$,the velocity attained by the rocket is $v \approx 0.69 \ v_e$.

Class 11 Physics Newton's Laws of Motion and Friction Variable Mass System and Rocket Problem

Difficult

$A$ stationary rocket of initial mass $m_0$ is expelling exhaust gas at a steady speed $v_e$ with respect to itself. There is no external force on the rocket. When its mass is reduced to $\frac{m_0}{2}$,the velocity attained by the rocket will be:

A
$2 \ v_e$
B
$v_e$
C
$0.69 \ v_e$
D
$0.5 \ v_e$

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Chapter

Newton's Laws of Motion and Friction

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Variable Mass System and Rocket Problem

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DifficultJEE MAIN 2025

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$A$ Diwali rocket of mass $100\,g$ lifts up with an acceleration of $5\,m/s^2$. If gases eject out of the rocket at a rate of $10\,g/s$,then find the velocity of ejection of gas with respect to the rocket in $m/s$.

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$A$ rocket of mass $5700 \,kg$ ejects mass at a constant rate of $15 \,kg/s$ with a constant speed of $12 \,km/s$. The acceleration of the rocket $1 \,minute$ after the blast is ........... $m/s^2$ $(g=10 \,m/s^2)$.

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$A$ balloon and its content having mass $M$ is moving up with an acceleration $a$. The mass that must be released from the content so that the balloon starts moving up with an acceleration $3a$ will be: (Take $g$ as acceleration due to gravity)

DifficultJEE MAIN 2025

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$A$ stationary rocket of initial mass $m_0$ is expelling exhaust gas at a steady speed $v_e$ with respect to itself. There is no external force on the rocket. When its mass is reduced to $\frac{m_0}{2}$,the velocity attained by the rocket will be:

Similar Questions

$A$ sand dropper drops sand of mass $m(t)$ on a conveyer belt at a rate proportional to the square root of speed $v$ of the belt,i.e.,$\frac{dm}{dt} \propto \sqrt{v}$. If $P$ is the power delivered to run the belt at constant speed,then which of the following relationships is true?

$A$ block of metal weighing $2 \, kg$ is resting on a frictionless plane. It is struck by a jet releasing water at a rate of $1 \, kg/s$ and at a speed of $5 \, m/s$. The initial acceleration of the block will be .......... $m/s^2$.

$A$ Diwali rocket of mass $100\,g$ lifts up with an acceleration of $5\,m/s^2$. If gases eject out of the rocket at a rate of $10\,g/s$,then find the velocity of ejection of gas with respect to the rocket in $m/s$.

$A$ rocket of mass $5700 \,kg$ ejects mass at a constant rate of $15 \,kg/s$ with a constant speed of $12 \,km/s$. The acceleration of the rocket $1 \,minute$ after the blast is ........... $m/s^2$ $(g=10 \,m/s^2)$.

$A$ balloon and its content having mass $M$ is moving up with an acceleration $a$. The mass that must be released from the content so that the balloon starts moving up with an acceleration $3a$ will be: (Take $g$ as acceleration due to gravity)

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