Answer
Verified
64.8k+ views
Hint Firstly, find mass of the rocket from given values of weight and acceleration due to gravity. Then, use it to find acceleration of the rocket from the given value of net force and calculated mass of the rocket. Then put acceleration of the rocket into the equation of motion: $v = u + at$, along with other given values of initial and final speeds, to get how much time will be required.
Complete step by step solution
We are given that the weight of the rocket is $W = 1.5 \times {10^4}$ N and acceleration due to gravity is $g = 10m/{s^2}$ .
We know that the weight of the rocket (W) is equal to acceleration due to gravity (g) times the mass of the rocket (m). $\therefore W = mg$
$ \Rightarrow m = \dfrac{W}{g}$
Putting in the values of W and g in the above equation,
$
\Rightarrow m = \dfrac{{1.5 \times {{10}^4}}}{{10}} \\
\Rightarrow m = 1.5 \times {10^3}kg \\
$
Hence, the mass of the rocket is $m = 1.5 \times {10^3}kg$.
Net force on the rocket (F) will be equal to acceleration of the rocket(a) times the mass of the rocket (m). We are given that the net force on the rocket is $F = 2.4 \times {10^4}$N.
$
\Rightarrow F = ma \\
\Rightarrow a = \dfrac{F}{m} \\
$
Substituting the values of net force on the rocket (F), which is given and mass of the rocket (m) that we calculated above in this expression for acceleration of the rocket gives us,
$
\Rightarrow a = \dfrac{{2.4 \times {{10}^4}N}}{{1.5 \times {{10}^3}kg}} \\
\Rightarrow a = 1.6 \times 10\dfrac{{kgm{s^{ - 2}}}}{{kg}} \\
\Rightarrow a = 16m{s^{ - 2}} \\
$
Now, we are given our initial speed of the rocket is 12 \[m/s\] and the final speed of the rocket must become 36 m/s or we can say $u = 12m/s$ and $v = 36m/s$ .
Using the equation of motion : $v = u + at$ and substituting all the known values,
\[
\Rightarrow 36 = 12 + \left( {16 \times t} \right) \\
\Rightarrow 36 - 12 = 16t \\
\Rightarrow 24 = 16t \\
\Rightarrow t = \dfrac{{24}}{{16}} \\
\Rightarrow t = 1.5\sec \\
\]
Therefore, the time required to increase the rocket’s speed from 12 \[m/s\] to 36 \[m/s\] as per the given parameters will be 1.5 seconds.
Hence, Option (C) is correct.
Note: An alternate short-cut method:
Take, Impulse = Force × Time
$
\Rightarrow mv - mu = Ft \\
\Rightarrow m(v - u) = Ft \\
\Rightarrow \dfrac{W}{g}(v - u) = Ft \\
\Rightarrow \dfrac{{W(v - u)}}{{gF}} = t \\
$
Putting in all the known values in the above equation yields,
$
\Rightarrow t = \dfrac{{1.5 \times {{10}^4} \times \left( {36 - 12} \right)}}{{10 \times 2.4 \times {{10}^4}}} \\
\Rightarrow t = \dfrac{{1.5 \times 24}}{{24}} \\
\Rightarrow t = 1.5\sec \\
$
Complete step by step solution
We are given that the weight of the rocket is $W = 1.5 \times {10^4}$ N and acceleration due to gravity is $g = 10m/{s^2}$ .
We know that the weight of the rocket (W) is equal to acceleration due to gravity (g) times the mass of the rocket (m). $\therefore W = mg$
$ \Rightarrow m = \dfrac{W}{g}$
Putting in the values of W and g in the above equation,
$
\Rightarrow m = \dfrac{{1.5 \times {{10}^4}}}{{10}} \\
\Rightarrow m = 1.5 \times {10^3}kg \\
$
Hence, the mass of the rocket is $m = 1.5 \times {10^3}kg$.
Net force on the rocket (F) will be equal to acceleration of the rocket(a) times the mass of the rocket (m). We are given that the net force on the rocket is $F = 2.4 \times {10^4}$N.
$
\Rightarrow F = ma \\
\Rightarrow a = \dfrac{F}{m} \\
$
Substituting the values of net force on the rocket (F), which is given and mass of the rocket (m) that we calculated above in this expression for acceleration of the rocket gives us,
$
\Rightarrow a = \dfrac{{2.4 \times {{10}^4}N}}{{1.5 \times {{10}^3}kg}} \\
\Rightarrow a = 1.6 \times 10\dfrac{{kgm{s^{ - 2}}}}{{kg}} \\
\Rightarrow a = 16m{s^{ - 2}} \\
$
Now, we are given our initial speed of the rocket is 12 \[m/s\] and the final speed of the rocket must become 36 m/s or we can say $u = 12m/s$ and $v = 36m/s$ .
Using the equation of motion : $v = u + at$ and substituting all the known values,
\[
\Rightarrow 36 = 12 + \left( {16 \times t} \right) \\
\Rightarrow 36 - 12 = 16t \\
\Rightarrow 24 = 16t \\
\Rightarrow t = \dfrac{{24}}{{16}} \\
\Rightarrow t = 1.5\sec \\
\]
Therefore, the time required to increase the rocket’s speed from 12 \[m/s\] to 36 \[m/s\] as per the given parameters will be 1.5 seconds.
Hence, Option (C) is correct.
Note: An alternate short-cut method:
Take, Impulse = Force × Time
$
\Rightarrow mv - mu = Ft \\
\Rightarrow m(v - u) = Ft \\
\Rightarrow \dfrac{W}{g}(v - u) = Ft \\
\Rightarrow \dfrac{{W(v - u)}}{{gF}} = t \\
$
Putting in all the known values in the above equation yields,
$
\Rightarrow t = \dfrac{{1.5 \times {{10}^4} \times \left( {36 - 12} \right)}}{{10 \times 2.4 \times {{10}^4}}} \\
\Rightarrow t = \dfrac{{1.5 \times 24}}{{24}} \\
\Rightarrow t = 1.5\sec \\
$
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
Other Pages
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
Electric field due to uniformly charged sphere class 12 physics JEE_Main
According to classical free electron theory A There class 11 physics JEE_Main
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main