How to convert acceleration as a funtion of position to a function of time?

In summary, the conversation discusses converting acceleration defined as a function of position, a(x), into a function of time, a(t). An example is given for the case a(x)= x/s², with the general solution for x and a(t) derived. The concept of using the work-energy theorem and solving for x(t) is also mentioned as a general method, with the caveat that t(x) must be invertible.
  • #1
WindScars
50
0
Suppose I have acceleration defined as a function of position, "a(x)". How to convert it into a function of time "a(t)"? Please give an example for the case a(x)= x/s²
 
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  • #2
not sure what s is supposed to be, if its a function of t or x, this won't work, but anything else is fine.

a(x)=x/s^2 = F/m

x'' - mx/s^2 = 0

has the general solution x = c*e^(sqrt(m)*t/s)+ d*e^(-sqrt(m)*t/s)

differentiating this twice gives you
a(t) = (mc/s^2)*e^(sqrt(m)*t/s) + (md/s^2)*e^(-sqrt(m)*t/s)

where c and d are constants you can find from boundary conditions or initial values
 
  • #3
That's correct except drop the "m", it doesn't belong. Also, there's a general way to do this-- write a(x) = v*dv/dx, use that to derive the work-energy theorem that the integral of a(x) over dx equals the change in v2/2. Solve that for v(x), and say t = integral of v(x) over dx. That gives you t(x), which you can invert to x(t), then plug into a(x(t)). This requires that t(x) be invertible, but it's the best you can do in general.
 

1. How can acceleration be expressed as a function of position?

Acceleration can be expressed as a function of position by taking the second derivative of the position function with respect to time. This means that the position function must be differentiated twice in order to obtain the acceleration function.

2. Why is it useful to convert acceleration as a function of position to a function of time?

Converting acceleration as a function of position to a function of time allows us to understand the rate of change of an object's velocity over time. This can help us analyze the motion of objects and make predictions about their future movements.

3. Can acceleration as a function of position and acceleration as a function of time be used interchangeably?

No, acceleration as a function of position and acceleration as a function of time are not interchangeable. While they both describe the acceleration of an object, they are expressed in different units and represent different relationships.

4. What is the difference between acceleration as a function of position and acceleration as a function of time?

The main difference between acceleration as a function of position and acceleration as a function of time is the independent variable. In acceleration as a function of position, the independent variable is position, while in acceleration as a function of time, the independent variable is time.

5. Are there any limitations to converting acceleration as a function of position to a function of time?

Yes, there are limitations to converting acceleration as a function of position to a function of time. This conversion assumes that the acceleration is constant over the entire range of positions, which may not always be the case. Additionally, this conversion may not be applicable for complex motion patterns or if the acceleration is changing over time.

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