Velocity as function of Displacement to Displacement as function of Time

In summary, the conversation discusses the measurement of velocities of a particle at varying displacements and the desire to find a function for displacement as a function of time. The differential equation dx/dt= 0.0002x^2 - 0.6484x + 885 is introduced, and the use of the chain rule and integration is mentioned as a possible solution. The conversation ends with a clarification of a typo.
  • #1
StephenSF8
3
0
I've measured velocities of a particle at varying displacements and characterized the velocity as [tex]V(x) = 0.0002x^2 - 0.6484x + 885[/tex].

You can see that I know velocity (V) as a function of displacement (x). Ultimately I want to end up with a function for displacement as a function of time (t). I imagine that somehow a chain rule is used to change the variables, but I'm having trouble figuring it out. The books I have glaze over the issue of non-constant acceleration...

Will somebody with more calculus experience help me out? Thanks.

Oh, and the initial conditions are x = 0, and so V(0) = 885.
 
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  • #2
Since v= dx/dt you have, effectively, the differential equation [itex]dx/dt= 0.0002x^2 - 0.6484x + 885[/itex] which you can rewrite
[tex]\frac{dv}{0.0002x^2 - 0.6484x + 885}= dt[/itex]
Factor the denominator and use "partial fractions" to integrate.
 
  • #3
Shouldn't that "dv" in the last term be a "dx" ?
 
  • #4
yes.
 

1. What is the formula for velocity as a function of displacement?

The formula for velocity as a function of displacement is v = Δd/Δt, where v represents velocity, Δd represents change in displacement, and Δt represents change in time.

2. How is displacement related to time in this function?

In this function, displacement is directly related to time. As time increases, displacement also increases. This relationship can be seen in the slope of the displacement vs. time graph, which represents the velocity at any given point in time.

3. How does velocity change as displacement increases?

As displacement increases, velocity can either increase, decrease, or remain constant depending on the direction of motion. If displacement and velocity are in the same direction, velocity will increase. If they are in opposite directions, velocity will decrease. If displacement is constant, velocity will remain constant.

4. Can velocity be negative in this function?

Yes, velocity can be negative in this function. A negative velocity indicates that an object is moving in the opposite direction of its initial displacement.

5. How does a change in displacement affect the velocity of an object?

A change in displacement directly affects the velocity of an object. If displacement increases, velocity will also increase, and vice versa. This is because displacement and velocity have a linear relationship in this function.

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