Rocket Mechanism

Q. A rocket of mass $20\text{kg}$ has $180\text{kg}$ fuel. The exhaust velocity of the fuel is $1.6\text{km/s}$. Calculate the minimum rate of ejection of fuel so that the rocket may rise from the ground. (Take $g=9.8{\text{m/s}}^2$)

1. $1.225\text{kg/s}$
2. $1.725\text{kg/s}$
3. $2.125\text{kg/s}$
4. $2.425\text{kg/s}$

Q. 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 $500\text{kg}$ ].

1. $20.4\text{m/s}$
2. $150.4\text{m/s}$
3. $30.4\text{m/s}$
4. $15.6\text{m/s}$

Q. $PSLV$ 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 $g=10{\text{m/s}}^2$]

1. $400\text{kg}$
2. $1000\text{kg}$
3. $500\text{kg}$
4. $200\text{kg}$

Q. A rocket with an initial mass of $M=2000\text{kg}$ is launched vertically upwards from rest, under gravity. The burnt matter is ejected vertically downwards with a speed of $4000\text{m/s}$ relative to the rocket. Velocity of the rocket after $60$ seconds is $2000\text{m/s}$. Find the mass of rocket gas ejected in $60$ seconds. [Take $g=10{\text{m/s}}^2$]

1. $420.62\text{kg}$
2. $955.86\text{kg}$
3. $590.62\text{kg}$
4. $500.92\text{kg}$

Q. $PSLV$ launches a rocket which is ejecting gas with a speed of $100\text{m/s}$ w.r.t the rocket and the rate of ejection of gas from the rocket is $400\text{kg/s}$. If the mass of the rocket is $2000\text{kg}$, calculate the initial acceleration of the rocket. [Take $g=10{\text{m/s}}^2$]

1. $10{\text{m/s}}^2$
2. $20{\text{m/s}}^2$
3. $5{\text{m/s}}^2$
4. $15{\text{m/s}}^2$

Q. 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

1. $109\text{km/s}$
2. $10.9\text{km/s}$
3. $109\text{m/s}$
4. $1.09\text{km/s}$

Q. $PSLV$ launches a rocket which is ejecting gas with a speed of $100\text{m/s}$ w.r.t the rocket and the rate of ejection of gas from the rocket is $400\text{kg/s}$. If the mass of the rocket is $2000\text{kg}$, calculate the initial acceleration of the rocket. [Take $g=10{\text{m/s}}^2$]

1. $10{\text{m/s}}^2$
2. $20{\text{m/s}}^2$
3. $5{\text{m/s}}^2$
4. $15{\text{m/s}}^2$

Q. In case of a variable mass system, a thrust force of magnitude $\left|V_r\frac{dm}{dt}\right|$ has to be applied on the system, whose mass is changing. Which of the following statements is/are correct?

1. Direction of thrust force is along $V_r$ if the mass is increasing.
2. Direction of thrust force is opposite to $V_r$ if the mass is increasing
3. Direction of thrust force is opposite to $V_r$ if the mass is decreasing.
4. Direction of thrust force is along $V_r$ if the mass is decreasing.