Animated Solution for Physics - Laws of Motion: The initial mass of a rocket is $1000\text{ kg}$. Calculate at what rate the fuel should be burnt, so that the rocket is given an acceleration of $20{\text{ ms}}^{-1}$. The gases come out at a relative speed of $500{\text{ ms}}^{-1}$ with respect to the rocket [Use, $g=10{\text{ m/s}}^2$]

Imagine a rocket standing on the launchpad, ready to defy gravity. How does it actually lift off? It doesn't push against the ground; instead, it relies on the principle of conservation of momentum. By violently expelling exhaust gases downwards, the rocket experiences an equal and opposite force upwards. This force is called thrust.

Let's break down the forces acting on our rocket. We have the upward thrust force, $F_{\text{thrust}}$, generated by the burning fuel, and the downward pull of gravity, which is the rocket's weight, $Mg$.

According to Newton's Second Law, the net force acting on an object is equal to its mass times its acceleration ($F_{\text{net}}=Ma$). Since the rocket is accelerating upwards, the net force is the difference between the thrust and the weight:

But how do we calculate the thrust itself? In a variable mass system like a rocket, thrust is the product of the relative velocity of the exhaust gases ($v_{\text{rel}}$) and the rate at which mass is being ejected ($\frac{dM}{dt}$):

Substituting this into our Newton's Second Law equation, we get the master equation for rocket motion:

Now, it's just a matter of plugging in the numbers. We know the initial mass $M=1000\text{ kg}$, the desired acceleration $a=20{\text{ m/s}}^2$, the relative velocity of the gases $v_{\text{rel}}=500\text{ m/s}$, and the acceleration due to gravity $g=10{\text{ m/s}}^2$.

Moving the weight term to the right side, we find the total force the thrust must provide:

Finally, dividing by the relative velocity gives us the required rate of fuel consumption:

So, the rocket must burn $60\text{ kg}$ of fuel every single second to achieve that initial acceleration. Notice that as the fuel burns, the rocket's mass $M$ will decrease. If the burn rate $\frac{dM}{dt}$ remains constant, the acceleration $a$ will continuously increase as the rocket climbs higher!