Velocity as function of Displacement to Displacement as function of Time

Click For Summary

Discussion Overview

The discussion revolves around the conversion of a velocity function defined in terms of displacement into a displacement function defined in terms of time. Participants explore the implications of non-constant acceleration and the application of calculus techniques to achieve this transformation.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant presents a velocity function V(x) = 0.0002x^2 - 0.6484x + 885 and seeks assistance in deriving displacement as a function of time.
  • Another participant identifies the relationship between velocity and displacement as a differential equation dx/dt = 0.0002x^2 - 0.6484x + 885 and suggests rewriting it for integration.
  • A question is raised regarding a potential error in notation, specifically whether "dv" should actually be "dx" in the context of the differential equation.
  • A subsequent reply confirms that the notation should indeed be corrected to "dx."

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct formulation of the differential equation, as a notation error is identified but not resolved in terms of its implications for the overall problem.

Contextual Notes

The discussion highlights the challenge of dealing with non-constant acceleration and the need for careful application of calculus techniques, particularly in the context of integrating differential equations.

Who May Find This Useful

This discussion may be of interest to those studying calculus, particularly in relation to motion and dynamics, as well as individuals seeking to understand the conversion between different mathematical representations of motion.

StephenSF8
Messages
3
Reaction score
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.
 
Physics news on Phys.org
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]<br /> Factor the denominator and use "partial fractions" to integrate.[/tex]
 
Shouldn't that "dv" in the last term be a "dx" ?
 
yes.
 

Similar threads

Undergrad Instantaneous velocity - displacement and distance
  • · Replies 3 ·
Replies
3
Views
2K
High School Kinematic equations ##\textbf{purely}## from graphs
  • · Replies 49 ·
2
Replies
49
Views
4K
Undergrad Simple harmonic motion equations as a function of time
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Work-Energy Theorem for Moving Block and Spring System?
  • · Replies 4 ·
Replies
4
Views
3K
High School In uniform acceleration, mean velocity ##\bar v = \frac{v_0+v}{2}##?
  • · Replies 23 ·
Replies
23
Views
3K
Undergrad Calculating distance from acceleration as function of speed
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Quick Question about Pendulum Graphs
  • · Replies 8 ·
Replies
8
Views
7K
Undergrad Viscosity Displacement Velocity Time relation
  • · Replies 13 ·
Replies
13
Views
3K
Undergrad Newton's second law and pressure wave propagation
  • · Replies 27 ·
Replies
27
Views
3K
High School Sign conventions in torque and non-uniform circular motion
  • · Replies 1 ·
Replies
1
Views
2K