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

• Cinitiator
In summary, the question is asking for the method to find the distance traveled by an object after being subjected to a force for a certain amount of time, given the force and mass functions, as well as initial conditions of velocity, acceleration, and distance. The suggested method involves finding the acceleration function by dividing the instantaneous force by the instantaneous mass and integrating it to find the velocity and distance functions. If the functions are not continuous, they can be split into continuous pieces and "bridged" with assumed constant velocity sections. Definite integration can be used for this purpose. However, the integrals may not always be defined, in which case the behavior of the functions must be interpreted in terms of physical laws.

## 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?

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.

gneill said:
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?

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

Sure.

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 guarantee that any of those functions will be "integrable" as an elementary function.

HallsofIvy said:
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 guarantee 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?

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Cinitiator said:
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.

gneill said:
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?

Cinitiator said:
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?

Cinitiator said:

## 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

## 1. How do force and mass affect distance traveled?

The distance traveled is directly proportional to the force applied and the mass of the object. This means that as the force increases, the distance traveled also increases, and as the mass increases, the distance traveled decreases.

## 2. How do we calculate the distance traveled using force and mass?

The distance traveled can be calculated by using the equation d = F/m, where d is the distance traveled, F is the force applied, and m is the mass of the object. This equation is derived from Newton's second law of motion.

## 3. Can the distance traveled be affected by other factors besides force and mass?

Yes, the distance traveled can also be affected by other factors such as friction, air resistance, and the surface the object is traveling on. These factors can either increase or decrease the distance traveled.

## 4. How can we measure the force and mass of an object to calculate the distance traveled?

The force applied to an object can be measured using a force sensor or a spring scale. The mass of an object can be measured using a balance or a scale. Once these values are obtained, they can be plugged into the equation d = F/m to calculate the distance traveled.

## 5. Is there a limit to how far an object can travel based on its force and mass?

Yes, there is a limit to how far an object can travel based on its force and mass. This is due to the law of conservation of energy, which states that energy cannot be created or destroyed, only transferred. Therefore, the initial force and mass of an object determine its maximum potential distance traveled.