Calculating Acceleration with Varying Mass: Tutorial & Exercises

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Discussion Overview

The discussion revolves around calculating acceleration in scenarios where mass is changing while thrust remains constant. Participants explore the implications of this relationship on displacement and velocity, particularly in the context of non-constant acceleration and the need for calculus in these calculations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant seeks to understand how acceleration changes with varying mass and constant thrust, expressing a desire for tutorials and exercises on the topic.
  • Another participant confirms the relationship a = T/m and notes that calculus is necessary for further analysis.
  • A participant questions the applicability of the standard equation for displacement under conditions of non-constant acceleration.
  • It is suggested that for non-constant acceleration, one must integrate acceleration over time to compute changes in velocity and position, leading to the need for double integration.
  • A reference to conservation of momentum is introduced, discussing the implications of external forces on momentum conservation and its derivatives.

Areas of Agreement / Disagreement

Participants generally agree on the need for calculus to address the problem of varying mass and its effects on acceleration, but there is no consensus on the specific methods for calculating displacement and velocity under these conditions.

Contextual Notes

The discussion highlights the limitations of standard equations of motion when acceleration is not constant and the need for integration techniques, which remain unresolved in terms of specific applications or examples.

Who May Find This Useful

This discussion may be useful for students or enthusiasts interested in advanced mechanics, particularly those dealing with variable mass systems and the mathematical techniques required for analysis.

mraptor
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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
 
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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.
 
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 ?
 
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
 
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.
 

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