Rocket Mechanism

Q. A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate $\frac{dM\left(t\right)}{dt}=b{v}^2\left(t\right)$, where $v\left(t\right)$ is its instantaneous velocity. The instantaneous acceleration of the satellite is

1. $-b{v}^3\left(t\right)$
2. $-\frac{b{v}^3}{M\left(t\right)}$
3. $-\frac{2b{v}^3}{M\left(t\right)}$
4. $-\frac{b{v}^3}{2M\left(t\right)}$

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 rocket with an initial mass of $1000\text{kg}$ is launched vertically upwards from rest under gravity. The rocket burns fuel at the rate of $10\text{kg}$ per second. The burnt matter is ejected vertically downwards with a speed of $2000\text{m/s}$ relative to the rocket. If burning ceases after one minute, the maximum velocity of the rocket is

1. $4000.6\text{m/s}$
2. $2500.4\text{m/s}$
3. $4500.2\text{m/s}$
4. $1232.6\text{m/s}$

Q. A rocket, set for vertical firing, weighs $50\text{kg}$ and contains $450\text{kg}$ of fuel. It can have maximum exhaust velocity of $2\text{km/s}$. What should be its minimum rate of fuel consumption, to just lift it off the launching pad ?

Take $g=9.8{\text{m/s}}^2$

1. $2.45\text{kg/s}$
2. $5.05\text{kg/s}$
3. $3.50\text{kg/s}$
4. $6.00\text{kg/s}$

Q. A rocket of mass $120\text{kg}$ has $280\text{kg}$ fuel. The relative exhaust velocity of the fuel is $1.8\text{km/s}$ and the rate of consumption of the fuel is $2\text{kg/s}$. Calculate the ultimate vertical speed gained by the rocket. (Take $g=9.8{\text{m/s}}^2$)

1. $3.54\text{km/s}$
2. $0\text{km/s}$
3. $1.79\text{km/s}$
4. $0.795\text{km/s}$

Q. A rocket of initial mass $m=2000\text{kg}$ is burning fuel at the rate of $20\text{kg/s}$ and ejecting burnt matter with a constant speed of $1000\text{m/s}$ w.r.t the rocket. Find the acceleration of the rocket at $t=10\text{sec}$.
[Take $g=10{\text{m/s}}^2$]

1. $0$
2. $0.55{\text{m/s}}^2$
3. $1.11{\text{m/s}}^2$
4. $2.3{\text{m/s}}^2$

Q. The exhaust velocity of the gases coming out of a rocket is $350\text{m/s}$ w.r.t the rocket. The rocket can burn fuel at a rate of $80\text{kg/s}$. For a successful launch, the desired acceleration of the rocket is $4{\text{m/s}}^2$. What is the initial total mass of this rocket? (Take $g=10{\text{m/s}}^2$)

1. $2000\text{kg}$
2. $3000\text{kg}$
3. $1500\text{kg}$
4. $2500\text{kg}$

Q. Find the mass $\left(m\right)$ of the rocket as a function of time, if it moves with a constant acceleration $a$ in the absence of external force. The gas escapes with a constant velocity $u$ relative to the rocket and its initial mass was $m_0$.

1. $m=m_0{e}^{-at/u}$
2. $m=m_0{e}^{at/u}$
3. $m=m_0{e}^{-t/u}$
4. $m=m_0{e}^{2at/u}$

Q. The figure shows the velocity - time graph of a ball of mass $20\text{g}$ moving along a straight line on a table. How much average force does the table exert on the ball to bring it to rest?

1. $0.02\text{N}$
2. $0.03\text{N}$
3. $0.04\text{N}$
4. $0.05\text{N}$

Q. If the force on a rocket, moving with a velocity $500\text{m/s}$ is $400\text{N}$, the rate of combustion of the fuel will be:

1. $10.8\text{kg/sec}$
2. $8\text{kg/sec}$
3. $0.8\text{kg/sec}$
4. $1.6\text{kg/sec}$

Q. A rocket of mass $40\text{kg}$ has $160\text{kg}$ fuel. The exhaust velocity of the fuel is $2\text{km/s}$. The rate of consumption of fuel is $4\text{kg/s}$. Calculate the ultimate vertical speed gained by the rocket. Assume that value of $g$ does not vary.
$(g=10{\text{m/s}}^2\text{and}\ln 5=1.609)$

1. $3218\text{m/s}$
2. $2818\text{m/s}$
3. $2418\text{m/s}$
4. $2018\text{m/s}$

Q. A spacecraft engine ejects mass at a rate of $30\text{kg/s}$ with an exhaust velocity of $3000\text{m/s}$. What is the thrust of the engine in vaccum?

1. $90,000\text{N}$
2. $900\text{N}$
3. $900000\text{N}$
4. $9000\text{N}$

Q. A rocket is in gravity free space. A scientist wants the acceleration of the rocket to be $2{\text{m/s}}^2$. Initial mass of the rocket is $1000\text{kg}$ and the speed of ejection of burnt fuel w.r.t the rocket is $500\text{m/s}$. Find the rate of consumption of fuel $\left(\text{in kg/s}\right)$, so that the rocket will have an acceleration of $2{\text{m/s}}^2.$

1. $2\text{kg/s}$
2. $4\text{kg/s}$
3. $1\text{kg/s}$
4. $3\text{kg/s}$

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.

Q. In the case of rocket propulsion, choose the correct option(s).

1. Momentum of system (rocket + fuel) always remains constant.
2. Newton's third law is applied.
3. If exhaust velocity and rate of burning of mass is kept constant, then acceleration of the rocket will go on increasing.
4. Newton's second law can be applied.

Q. A rocket with an initial mass $m=2000\text{kg}$ is launched vertically upwards. The rocket burns fuel at the rate of $20\text{kg/s}$. The burnt material is ejected vertically downwards with a speed of $4000\text{m/s}$ relative to the rocket. If burning stops after $30$ seconds, find the maximum velocity of the rocket. (Take $g=10{\text{m/s}}^2$ )

1. $0\text{m/s}$
2. $4515.9\text{m/s}$
3. $1126.7\text{m/s}$
4. $300\text{m/s}$

Q. Mass of spacecraft varies with time as $m=5000e^{-t}\text{kg}$. Find the rate of fuel ejection after $5\text{s}$. (Take $e=2.718$)

1. $23.7\text{kg/s}$
2. $33.7\text{kg/s}$
3. $337\text{kg/s}$
4. $237\text{kg/s}$

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

Q. A rocket of mass $3500\text{kg}$ has to achieve a velocity of $1000\text{m/s}$ over a launch time of $10$ minutes. The nozzle of the rocket is designed to let out exhaust gases at a velocity of $300\text{m/s}$ w.r.t the rocket. What is the amount of fuel that is consumed during the launch?

1. $90000\text{kg}$
2. $88666.66\text{kg}$
3. $81660\text{kg}$
4. $66666.66\text{kg}$

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. A rocket of initial mass $3000\text{kg}$ is burning fuel at the rate of $30\text{kg/s}$. Find the mass of the rocket after $10\text{s}$.

1. $2700\text{kg}$
2. $3300\text{kg}$
3. $2000\text{kg}$
4. $0\text{kg}$

Q. A rocket with an initial mass of $M=2000\text{kg}$ is launched vertically upwards from rest, under gravity. The rocket burns fuel at the rate of $20\text{kg/s}$. 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. $955.86\text{kg}$
2. $500.92\text{kg}$
3. $420.62\text{kg}$
4. $590.62\text{kg}$

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

Q. In the arrangements shown in figure masses of each ball is $1kg$ and mass of trolley is $4kg$. In the figure shell of mass $1kg$ moving horizontally with velocity $v=6m/sec$ collides with the ball and get stuck to it then its maximum deflection (in degree) of the thread (length $1.5m$) with verticle is:

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. With what acceleration $a$ should the box of figure be moving upwards so that the block of mass $m$ exerts a force $\frac{7Mg}{4}$ on the floor of the box ?

1. $\frac{g}{4}$
2. $\frac{g}{2}$
3. $\frac{3g}{4}$
4. $4g$

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 (in ${\text{m/s}}^2$ ). [Take $g=10{\text{m/s}}^2$]

Q. The exhaust velocity of the gases coming out of a rocket is $350\text{m/s}$ w.r.t the rocket. The rocket can burn fuel at a rate of $80\text{kg/s}$. For a successful launch, the desired acceleration of the rocket is $4{\text{m/s}}^2$. What is the initial total mass of this rocket? (Take $g=10{\text{m/s}}^2$)

1. $3000\text{kg}$
2. $2000\text{kg}$
3. $1500\text{kg}$
4. $2500\text{kg}$

Q. A spacecraft engine ejects mass at a rate of $30\text{kg/s}$ with an exhaust velocity of $3000\text{m/s}$. What is the thrust of the engine in vaccum?

1. $90,000\text{N}$
2. $9000\text{N}$
3. $900000\text{N}$
4. $900\text{N}$

Q. A rocket has a mass of $100kg$. $90$ %of this is fuel. It ejects fuel vapours at the rate of $1$ kg/sec with a velocity of $500m/sec$ relative to the rocket. It is supposed that the rocket is outside the gravitational field.The initial upthrust on the rocket when it just starts moving upwards is (in $N$)

1. $500$
   
2. $0$
   
3. $200$
   
4. $300$