A rocket of initial mass $M=1000\text{ kg}$ ejects gas at a rate of $\frac{dM}{dt}=20\text{ kg/s}$ with a velocity of $u=500\text{ m/s}$ relative to the rocket. What is the initial acceleration $a$ of the rocket? (Take $g=10{\text{ m/s}}^2$)

Solution

1. Identify the given values: initial mass $M=1000\text{ kg}$, ejection rate $\frac{dM}{dt}=20\text{ kg/s}$, relative velocity $u=500\text{ m/s}$, and gravity $g=10{\text{ m/s}}^2$.
2. Calculate the thrust force using the formula $$\begin{array}{rl}{F}_{\text{thrust}}& =u\frac{dM}{dt}\\ & =500\times 20\\ & =10000\text{ N}\end{array}$$.
3. Calculate the weight of the rocket using $$\begin{array}{rl}W& =Mg\\ & =1000\times 10\\ & =10000\text{ N}\end{array}$$.
4. Determine the net force $$\begin{array}{rl}{F}_{\text{net}}& =F_{\text{thrust}}-W\\ & =10000\text{ N}-10000\text{ N}\\ & =0\text{ N}\end{array}$$.
5. Since $F_{\text{net}}=Ma$, the initial acceleration is $a=0{\text{ m/s}}^2$.
6. Hence the answer is (A).