What is v(dv/dx) and how does it relate to acceleration?

In summary, the conversation discusses the use of the chain rule to model acceleration as ##v(dv/dx)## in mechanics. This is related to the concept of material derivative and can be used to model motion with air resistance and in finding the time it takes for two masses to come together under gravity.
  • #1
Big-Daddy
343
1
My mechanics syllabus suggests that we can model acceleration as ##v(dv/dx)## but what exactly does this mean? Can you give me (or link me to) some explanations and/or sample problems with worked solutions? In my experience so far we should always be using ##dv/dt## for ##a##.
 
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  • #2
Consider what [itex]v[/itex] actually is in one dimension:
[tex]v = \dfrac{dx}{dt}.[/tex]
So in using the chain rule,
[tex]v\dfrac{dv}{dx} = \dfrac{dx}{dt}\dfrac{dv}{dx} = \dfrac{dv}{dt} = a.[/tex]
Of course, you could make the same argument with any spatial coordinate. This is intimately related to what is commonly called the material derivative.
 
  • #3
boneh3ad said:
Consider what [itex]v[/itex] actually is in one dimension:
[tex]v = \dfrac{dx}{dt}.[/tex]
So in using the chain rule,
[tex]v\dfrac{dv}{dx} = \dfrac{dx}{dt}\dfrac{dv}{dx} = \dfrac{dv}{dt} = a.[/tex]
Of course, you could make the same argument with any spatial coordinate. This is intimately related to what is commonly called the material derivative.

Thanks.

What would be the usefulness of something like this? Could you, for example, model motion with air resistance using it? (This is impossible with ##dv/dt## calculus since your balanced expression for acceleration would depend on -kv2, your air resistance, but then ##v## depends on acceleration, meaning the model cannot incorporate air resistance.)
 
  • #4
Suppose you have a stream of some fluid and you know the velocity field. You would like to know the acceleration of (and hence the forces on) some bit of fluid as it passes by a certain point. This formulation is useful.
 
  • #5
It's useful when separating differential equations like the following:

[tex]a=\frac{k}{r^2}[/tex]

So you can integrate:

[tex]vdv=\frac{k}{r^2}dr[/tex]

This is used, for example, with finding the time it takes for two masses to come together under gravity.
 

FAQ: What is v(dv/dx) and how does it relate to acceleration?

What does "V*dv/dx" represent in terms of acceleration?

"V*dv/dx" represents the rate of change of velocity with respect to distance, or the second derivative of position with respect to time. This is commonly referred to as acceleration, which is a measure of how quickly an object's velocity is changing.

How is "V*dv/dx" calculated?

The calculation of "V*dv/dx" involves taking the derivative of velocity with respect to distance. This can be done using the chain rule, where "v" represents velocity and "x" represents distance.

What is the difference between "V*dv/dx" and "a" in terms of acceleration?

"V*dv/dx" and "a" both represent acceleration, but they are different ways of expressing it. "V*dv/dx" is the second derivative of position with respect to time, while "a" is the rate of change of velocity with respect to time. Essentially, "V*dv/dx" is a more specific and precise way of measuring acceleration.

How does "V*dv/dx" relate to Newton's second law of motion?

Newton's second law of motion states that the net force acting on an object is equal to its mass multiplied by its acceleration (F=ma). The "V*dv/dx" term represents the acceleration of an object, and when multiplied by the mass of the object, gives the net force acting on it. Therefore, "V*dv/dx" is a crucial component in understanding and applying Newton's second law of motion.

What are some real-life applications of "V*dv/dx" for acceleration?

"V*dv/dx" has many practical applications in the study of motion and mechanics. For example, it is used in engineering and design to determine the acceleration and forces acting on structures or vehicles. It is also used in physics and astronomy to study the motion of objects in space. Additionally, "V*dv/dx" is important in fields such as sports science, where it can be used to analyze and improve the performance of athletes.

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