Velocity from acceleration if acceleration is a function of space

In summary, the conversation discusses the concept of acceleration and how it can be a function of either time or position. The chain rule is used to show the relationship between acceleration as a function of position and velocity as a function of time. It is also mentioned that it may not always be possible to find equations for acceleration, velocity, and position, and in those cases, numerical integration is necessary.
  • #1
fisico30
374
0
Hello Forum,

usually the acceleration a is a function of time: a(t)= d v(t) /dt

to find v(t) se simply integrate v(t)= integral a(t) dt

What if the acceleration was a function of space, i.e. a(x)?

what would we get by doing integral a(x) dx? The velocity as a function of space, v(x)?

but a(x) is not defined to be v(x)/dx or is it? maybe some chain rule is involved..

thanks,

fisico30
 
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  • #2
fisico30 said:
What would we get by doing integral a(x) dx? ... chain rule ...
You use chain rule

a(x) = dv/dt = f(x)

multiply dv/dt by dx/dx:

(dv/dt)(dx/dx) = (dx/dt)(dv/dx) = v dv/dx = f(x)

multiply both sides by dx

v dv = f(x) dx

If f(x) can be integrated and the integral of f(x) = g(x), then

1/2 v2 = g(x) + c

This results in an equation that relates velocity and position.

v = sqrt(2 (g(x) + c))

To get time versus position, you start with

v = dx/dt = sqrt(2 (g(x) + c))

and then integrate

dx/(sqrt(2 (g(x) + c))) = dt

assuming h(x) is the integral of dx/(sqrt(2 (g(x) + c))), you get

t = h(x) + d

where d is constant of integration. It may be possible to solve this equation to get x = some function of t.

One simple case is a(x) = -x, which results in x(t) = e(sin(t+f)), where e and f are constants.
 
Last edited:
  • #3
Thanks rgcldr!

great explanation and example!

fisico30
 
  • #4
Hi rcgldr,

one question:

when you get v dv = f(x) dx, is v a function of t or of x, i.e. is v equal to v(t) or v(x)?

It looks like it would be a function of x, v(x), since 1/2 v^2 = g(x) + c results in v(x) = sqrt(2 (g(x) + c)).

I am confused because when we write v=dx/dt I always assume that v must be a function of time t, i.e. v(t).
When in your first step you write a(x) = dv/dt I tend to think that the v must be function of t since dv/dt is a derivative with respect to time.

So which one is correct? a(x)= dv(x)/dt or a(t)=dv(t)/dt. Acceleration is always the time derivative of the velocity vector but the velocity vector can be a function of any dependent variable: v(t), v(x), v(F), where F is force, etc...

Thanks,
fisico30
 
  • #5
I guess my point is: if a function, like a, is defined as a time derivative of another function, dv/dt, does the differentiated function need to be a function if time?
 
  • #6
fisico30 said:
when you get v dv = f(x) dx, is v a function of t or of x, i.e. is v equal to v(t) or v(x)? It looks like it would be a function of x, v(x), since 1/2 v^2 = g(x) + c results in v(x) = sqrt(2 (g(x) + c)).
At this point, it's just a relationship betwen v and x.

I am confused because when we write v=dx/dt I always assume that v must be a function of time t, i.e. v(t).
It is, but I was only able to take advantage of that in the last step where I find t as a function of x.

When in your first step you write a(x) = dv/dt I tend to think that the v must be function of t since dv/dt is a derivative with respect to time.
True, but I used chain rule to get rid of the dt and end up with v dv = f(x) dx.

So which one is correct? a(x)= dv(x)/dt or a(t)=dv(t)/dt.
Both are correct, but if acceleration is defined as a function of x, then it may not be possible to find an equation for acceleration as a function of time. In the simple example I gave, a(x) = -x, you will be able to find both a(x) and a(t). I ended up finding x(t) = e sin(t + f). You can take the derivative of this to find v(t), and the derivative of v(t) to find a(t), so a(x) = -x, and a(t) = -e sin(t+f). For other situations, you may not be able to solve for x(t), v(t), or a(t), in which case numerical integration will be required.
 
  • #7
So it is mathematically legal and ok to write a derivative of a function even if
the independent variable a of the function and the differentiation variable b are not the same: f(a) and db to create df(a)/db...

I guess that is all the chain rule is about...
 

1. What is the relationship between velocity and acceleration if acceleration is a function of space?

Velocity and acceleration are related by the fundamental equation of motion, which states that velocity is equal to the integral of acceleration with respect to time. However, when acceleration is a function of space, this equation becomes more complex and can vary based on the specific function and its derivatives. In general, the velocity will still be affected by the acceleration, but the exact relationship may require more advanced mathematical analysis.

2. How does space affect the acceleration and velocity of an object?

In classical mechanics, space does not directly affect the acceleration or velocity of an object. However, if acceleration is a function of space, then space can indirectly impact these quantities. For example, if the acceleration is greater in one region of space compared to another, the velocity of an object may change accordingly.

3. Can acceleration be a function of both time and space?

Yes, acceleration can be a function of both time and space. In fact, most real-world situations involve acceleration that changes over both time and space. This can make predicting and analyzing an object's velocity more complex, as both factors must be taken into account.

4. How do you calculate velocity when acceleration is a function of space?

To calculate velocity when acceleration is a function of space, you will need to use calculus and integration. First, determine the function for acceleration as a function of space. Then, take the integral of this function with respect to time to find the velocity function. The velocity at a specific point in space can then be found by plugging in the corresponding value for time.

5. Can objects have different velocities at the same point in space?

Yes, objects can have different velocities at the same point in space if they have different histories of acceleration leading up to that point. For example, two objects may start at rest and then accelerate differently, resulting in different velocities at the same point in space. However, if the acceleration is constant, the velocities of the objects will eventually equalize at the same point in space.

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