Question

A rocket of mass 1000kg uses 5kg of fuel and oxygen to produce exhaust gases ejected at 500m/s calculate the increase its velocity?​

Answer 1

Answer:

Approximately $\rm 2.5\; m \cdot s^{-1}$.

Explanation:

Let the increase in the rocket's velocity be $\Delta v$. Let $v_0$ represent the initial velocity of the rocket. Note that for this question, the exact value of  $v_0$ doesn't really matter.

The momentum of an object is equal to its mass times its velocity.

• Mass of the rocket with the 5 kg of fuel: $1000$.
• Initial velocity of the rocket and the fuel: $v_0$.
• Hence the initial momentum of the rocket: $1000\,v_0$.

• Mass of the rocket without that 5 kg of fuel: $1000 - 5 = 995$.
• Final velocity of the rocket: $v_0 + \Delta v$.
• Hence the final momentum of the rocket: $995\,(v_0 + \Delta v)$.

• Mass of the 5 kg of fuel: $5$.
• Final velocity of the fuel: $v_0 - 500$ (assuming that the the 500 m/s in the question takes the rocket as its reference.)
• Hence the final momentum of the fuel: $5\,(v_0 - 500)$.

Momentum is conserved in an isolated system like the rocket and its fuel. That is:

Sum of initial momentum = Sum of final momentum.

$1000\,v_0 = 995\,(v_0 + \Delta v) + 5\,(v_0 - 500)$.

Note that $1000\, v_0$ appears on both sides of the equation. These two terms could hence be eliminated.

$0 = 995\, \Delta v - 5\times 500$.

$\displaystyle \Delta v = \frac{5}{995}\times 500 \approx \rm 2.5\; m \cdot s^{-1}$.

Hence, the velocity of the rocket increased by around 2.5 m/s.