Question

A $1000kg$ rocket is fired from the surface of the earth so that its exhaust speed is $1000m/s$ relative to the rocket. If the initial acceleration of the rocket is $5m/s^2$ then the mass of fuel burning per second is
(Take, $g=10m/s^2$)

• A. $15kg/s$
• B. $5kg/s$
• C. $10kg/s$
• D. $20kg/s$

Answer 1

The correct option is A

$15kg/s$

Step 1: Given Data

Initial mass of the rocket $M=1000kg$

Speed of the exhaust $v=1000m/s$

The initial acceleration of the rocket $a=5m/s^2$

Acceleration due to gravity $g=10m/s^2$

Step 2: Establish the Equation

We know that force is equal to the product of mass and acceleration.

Force is also equal to the rate of change of momentum $p$.

$\therefore M\left(g+a\right)=\frac{dp}{dt}$

We know that momentum $p=mv$

$\therefore M\left(g+a\right)=\frac{d}{dt}\left(mv\right)$

In this case, the velocity of the exhaust remains constant and the mass is changing with time.

$\therefore M\left(g+a\right)=v\frac{dm}{dt}$

$\Rightarrow \frac{dm}{dt}=\frac{M}{v}\left(g+a\right)$

Step 3: Calculate the rate of change in mass

Upon substituting the values we get,

$\frac{dm}{dt}=\frac{1000}{1000}\left(10+5\right)$

$=15kg/s$

Hence, the correct answer is option (A).