Calculating Acceleration with Varying Mass: Tutorial & Exercises

[mraptor]

hi,

I want to calculate how is acceleration changing if I have changing mass, but constant trust i.e. :

T = m*a

a = T / m

(I know it has to be calculus).
Then again I also wan't to be able to calculate displacement and velocities etc..
Trying to find somewhere on the internet a tutorial on equations of motion when the acceleration is varying.. but most of the time I find equations for constant-acceleration.
Do you have a good tutorial ? (don't point me to wikipedia, it is good as reference but not as tutorial)
I would like also to have some simple Exersises, so I can figure out how it is done in general.

thank you

 

[jbriggs444]

You are on the right track. a = T/m and it takes calculus.

The derivative of 1/m with respect to m is -1/m²

So for constant T, the derivative of T/m with respect to m is -T/m²

The minus sign indicates that as m increases the quotient T/m decreases.

 

[mraptor]

Nice.. ok now how can I calculate displacement or time taken to cross specific distans having this acceleration...
I suppose I can't use :

d = x + v*t + 1/2 a*t^2

because this is only valid for constant acceleration ?

 

[jbriggs444]

You are correct. For a non-constant acceleration instead of computing the change in velocity by simply multiplying acceleration by time, you have to compute it by integrating acceleration over time using calculus.

Similarly, for a non-constant velocity you compute change in position by integrating velocity over time rather than simply multiplying velocity by time.

You end up with a double integral.

The first integration to compute velocity as a function of time results in:

http://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation

 

[HallsofIvy]

Conservation of momentum, mv, leads to (mv)'= m'v+ mv'= 0 (the ' indicates the derivative) if there is no external force. If there is a force, then we do not have conservartion of momentum but have (mv)'= m'v+ mv'= F.