I'm new to rocket propulsion elements and I'm reading Rocket Propulsion Elements by George P. Sutton, 4th Edition.
In the second chapter, it introduced me the following expression,
$$I_{s}=\frac{F}{\dot{w}}=\frac{c}{g}$$
$I_{s}: \text{Specific impulse}$
$F: \text{Thrust}$
$\dot{w}: \text{Weight flow rate}$
$c: \text{Effective exhaust velocity}$
$g: \text{Gravitational constant}$
Now, I created an example myself. I considered that I have an amateur rocket engine that contains $1kg$ of propellant and I ignite this amateur rocket engine on the ground of the Earth (so, $g$ is basically $~9.8m/s^2$). So the initial weight of the propellant is $w_{i}=m_{i}g=1\times 9.8=9.8 kg.m/s^2$. I consider that the burn time of the propellant is $0.3$ seconds. So, the average weight flow rate is,
$$\frac{\Delta{w}}{\Delta{t}}=\frac{w_{f}-w_{i}}{t_{f}-t_{i}}=\frac{0-9.8}{0.3}=32,66kg.m/s^3$$
I give $F=50N$, so I can find the $I_{s}$ of my amateur rocket engine.
$$I_{s}=\frac{F}{\Delta{w}/ \Delta{t}}=\frac{50}{32,66}=0,153s$$
Now, I want to find out the effective velocity of my amateur rocket engine,
$$I_{s}=\frac{c}{g}\Rightarrow c=I_{s}g=0,153\times 9.8=1,5 m/s$$
Are my calculation correct and logic correct?