# How to get the distance traveled from the force and mass functions?

## Homework Statement

I have two functions:
F(t) - where F(t) is the force at a given time t
m(t) - where m(t) is the mass of the object in question at a given time t

Let's say that some force (in a thrust form) is applied to the object for "b" seconds. The function F(t) specifies in what manner.

How can I get the distance traveled by the object after "b" seconds, if we know that the velocity, acceleration and distance traveled are all 0 at t = 0?

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## The Attempt at a Solution

I've tried using an analogy of the Riemann sum (diving each instantaneous force by each instantaneous mass and summing everything for an acceleration-time function), and it turned out to be too tedious and imprecise to be applied practically.

Anyone?

gneill
Mentor
You can write an expression for the acceleration with respect to time from the given force and mass functions. Integrate to find velocity. Integrate again to find distance.

You can write an expression for the acceleration with respect to time from the given force and mass functions. Integrate to find velocity. Integrate again to find distance.

Thanks for your feedback. Can one just successively integrate the force function dived by the mass function to get the distance function?

gneill
Mentor
Thanks for your feedback. Can one just successively integrate the force function dived by the mass function to get the distance function?

Sure.

HallsofIvy
Homework Helper
m(t)dv/dt= f(t) so that dv= (f(t)/m(t))dt and you can integrate that. Once you have found v(t), you can use dx/dt= v(t) and integrate dx= v(t)dt to find the distance function. Of course, there is no guarentee that any of those functions will be "integrable" as an elementary function.

m(t)dv/dt= f(t) so that dv= (f(t)/m(t))dt and you can integrate that. Once you have found v(t), you can use dx/dt= v(t) and integrate dx= v(t)dt to find the distance function. Of course, there is no guarentee that any of those functions will be "integrable" as an elementary function.

But what if the F(t) and m(t) functions aren't continuous, and are only continuous on the interval of [0, b]? It's easy to do the calculations when the function is completely continuous. But how to do it if it's continuous only over [0, b], and we want to know the distance traveled at b?

Last edited:
gneill
Mentor
But what if the F(t) and m(t) functions aren't continuous, and are only continuous on the interval of [0, b]? It's easy to do the calculations when the function is completely continuous. But how to do it if it's continuous only over [0, b], and we want to know the distance traveled at b?

If they're continuous over [0,b] and you want the distance at b, I don't see the problem since the integrals will be defined over the domain.

If the functions are not continuous then it is up to you to interpret their behavior in terms of physical laws and deal with the implications. This might, for example, mean splitting the domain of integration into continuous pieces and "bridging" the gaps with assumed constant velocity sections.

If they're continuous over [0,b] and you want the distance at b, I don't see the problem since the integrals will be defined over the domain.

If the functions are not continuous then it is up to you to interpret their behavior in terms of physical laws and deal with the implications. This might, for example, mean splitting the domain of integration into continuous pieces and "bridging" the gaps with assumed constant velocity sections.

Can one use definite integration for that?

gneill
Mentor
Can one use definite integration for that?

Sure. Any integration is just a sum. A sum can be split into chunks and added separately. If some physics occurs between the parts represented by the integrations, then the integrations just become terms in an overall equation of motion where you stick other terms to fill in the "spaces".

Do you have some particular F(t) and M(t) in mind which is raising these concerns?

## Homework Statement

I have two functions: F(t) - where F(t) is the force at a given time t m(t) - where m(t) is the mass of the object in question at a given time t Let's say that some force (in a thrust form) is applied to the object for "b" seconds. The function F(t) specifies in what manner. How can I get the distance traveled by the object after "b" seconds, if we know that the velocity, acceleration and distance traveled are all 0 at t = 0?

-

## The Attempt at a Solution

I've tried using an analogy of the Riemann sum (diving each instantaneous force by each instantaneous mass and summing everything for an acceleration-time function), and it turned out to be too tedious and imprecise to be applied practically.[/QU