Answer
Verified
381.9k+ views
Hint:You can solve this question quite easily by using the fact that since there are no external forces, we can imply Newton's second law of motion and find a relation between the derivative of mass and acceleration. In other words, since net external force is absent which means rocket is propelled by relative change of momentum of exhaust gases with respect to rocket. Hence net force is equal to the rate of change of momentum of exhaust.
Complete step by step answer:
We will proceed with the solution exactly as told in the hint section of the solution to the question.
We’ve been given in the question that initial mass is ${m_o}$
Acceleration is constant and has a value of $a$ which is nothing but derivative of speed with respect to time, or $\dfrac{{dv}}{{dt}}$
The constant velocity with respect to the rocket with which the exhaust gases are escaping the rocket is given as $u$
Using Newton’s second law, we can write:
$F = \dfrac{{d\left( {mv} \right)}}{{dt}}$
Using the product rule of differentiation:
$F = m\dfrac{{dv}}{{dt}} + v\dfrac{{dm}}{{dt}}$.............................(i)
In the given question, it is given that:
$\dfrac{{dv}}{{dt}} = a$
$F = 0$ and,
$v = u$
Substituting the above-mentioned results in the equation (i):
$0 = ma + u\dfrac{{dm}}{{dt}}$
After transposing, we get:
$\dfrac{{dm}}{m} = - \dfrac{a}{u}dt$
Now, we can see that to reach the answer, we simply need to integrate the equation with proper limits.
At any given time, the value of time has gone from $0$ to $t$ and the value of mass has gone from ${m_o}$ to $m$. Now, integrating the found equation with the same limits, we get:
$\int\limits_{{m_o}}^m {\dfrac{{dm}}{m}} = \int\limits_0^t { - \dfrac{a}{u}dt} $
After integrating, we get:
$\ln \left( {\dfrac{m}{{{m_o}}}} \right) = - \dfrac{{at}}{u}$
After we transpose this further, we get the resulting equation as:
$m = {m_o}{e^{ - \dfrac{{at}}{u}}}$
And this is the equation of mass as a function of time as asked to us by the question.
Note: Many students take the limit as wrong as they don’t understand how mass can be reduced, we need to see that in the equations, the mass that has been talked about is not just the mass of the body of the rocket but the mass of the whole system, body of rocket and fuel combined.
Complete step by step answer:
We will proceed with the solution exactly as told in the hint section of the solution to the question.
We’ve been given in the question that initial mass is ${m_o}$
Acceleration is constant and has a value of $a$ which is nothing but derivative of speed with respect to time, or $\dfrac{{dv}}{{dt}}$
The constant velocity with respect to the rocket with which the exhaust gases are escaping the rocket is given as $u$
Using Newton’s second law, we can write:
$F = \dfrac{{d\left( {mv} \right)}}{{dt}}$
Using the product rule of differentiation:
$F = m\dfrac{{dv}}{{dt}} + v\dfrac{{dm}}{{dt}}$.............................(i)
In the given question, it is given that:
$\dfrac{{dv}}{{dt}} = a$
$F = 0$ and,
$v = u$
Substituting the above-mentioned results in the equation (i):
$0 = ma + u\dfrac{{dm}}{{dt}}$
After transposing, we get:
$\dfrac{{dm}}{m} = - \dfrac{a}{u}dt$
Now, we can see that to reach the answer, we simply need to integrate the equation with proper limits.
At any given time, the value of time has gone from $0$ to $t$ and the value of mass has gone from ${m_o}$ to $m$. Now, integrating the found equation with the same limits, we get:
$\int\limits_{{m_o}}^m {\dfrac{{dm}}{m}} = \int\limits_0^t { - \dfrac{a}{u}dt} $
After integrating, we get:
$\ln \left( {\dfrac{m}{{{m_o}}}} \right) = - \dfrac{{at}}{u}$
After we transpose this further, we get the resulting equation as:
$m = {m_o}{e^{ - \dfrac{{at}}{u}}}$
And this is the equation of mass as a function of time as asked to us by the question.
Note: Many students take the limit as wrong as they don’t understand how mass can be reduced, we need to see that in the equations, the mass that has been talked about is not just the mass of the body of the rocket but the mass of the whole system, body of rocket and fuel combined.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Trending doubts
State the differences between manure and fertilize class 8 biology CBSE
Why are xylem and phloem called complex tissues aBoth class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
What would happen if plasma membrane ruptures or breaks class 11 biology CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What precautions do you take while observing the nucleus class 11 biology CBSE
What would happen to the life of a cell if there was class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE