# I Rocket Example

1. Jul 26, 2017

### DonDiablo

Hi - I just thought of a (relatively) simple example: Here is the problem I can't solve due to my disability to integrate the resulting equation:

I thought about a rocket that gets accelerated by a constant force F... Since the rocket is burning fuel and therefor losing mass at a conatnt rate its acceleration is not constant. Its mass is given by m=m0 - mL*t - with m0 being its mass at the start of the operation and mL being the rate at which it is losing weight (being constant). Since the force with which the rocket is accelerated is constant I get the following equation:

F= (m0-(m/T)*t)*a now i want to form that so I get "a" which I want then to integrate after t to get a formular for the rockets velocity! a=F/(m0-(m/T)*t)! This is the formular I dont know how to integrate! Help would be greatly appreciated! Lg Don

2. Jul 26, 2017

### NFuller

$$a=\frac{dv}{dt}=\frac{F}{m_{0}-m_{L}t}$$
$$dv=\frac{Fdt}{m_{0}-m_{L}t}$$
$$\int dv=\int\frac{Fdt}{m_{0}-m_{L}t}$$
Using a u-substitution $u=1-m_{L}t/m_{0}$,
$$\int dv=-\frac{F}{m_{L}}\int\frac{du}{u}$$
$$v=-\frac{F}{m_{L}}\text{ln}(u)+C$$
$$v=-\frac{F}{m_{L}}\text{ln}\left(1-\frac{m_{L}}{m_{0}}t\right)+v_{0}$$

Last edited: Jul 26, 2017
3. Jul 27, 2017

### DonDiablo

Thanks a lot! I know that this is just a standard example but it still amazes me hoe you found the substitution! Wouldn't have come there so easy! Lg Don