# How to prove that position by velocity is a constant vector

• heenac2
In summary, the conversation discusses proving two equations for a harmonic function and how to finish the proof. The equations involve acceleration, velocity, and position and are true for all simple harmonic oscillators. The conversation also mentions using relationships, multiplying the starting equation by v, and integrating to derive the result for one of the equations.
heenac2
[Note from mentor: this thread was originally posted in a non-homework forum, therefore it does not use the homework template.]

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So I've been asked to prove that in a harmonic function where

a(t)+w2r(t)=0

that

(1) v(t).v(t)+w2r(t).r(t)=constant scalar

and

(2) r(t).v(t)=constant vector

where a(t)=acceleration, v(t)=velocity, r(t)=position

By deriving (1) I found that

2[a(t)+w2r(t)].v(t)=0 because a(t)+w2r(t)=0

By deriving (2) I get

v(t).v(t)+r(t)a(t)= v(t).v(t)+r(t)[-w2r(t)] because a(t)=-w2r(t)

How do I finish this?

Can anyone please explain what the point of this proof is?

Thanks!

Last edited by a moderator:
What is v(t) in terms of the time derivative of r(t)? What is a(t) in terms of the time derivative of v(t)?

Chet

heenac2
r⋅v isn't a vector. Is that perhaps supposed to be a cross product?

I don't think r⋅v is constant.

heenac2
Just solve the ODE, you'll get values of r and a, verify if these hold when plugging them in eq 1 and 2, Cheers!

heenac2
Chestermiller said:
What is v(t) in terms of the time derivative of r(t)? What is a(t) in terms of the time derivative of v(t)?

Chet

It doesn't actually v(t) in terms of r(t) or a(t) in terms of v(t). I'm meant to show that it's true for all simple harmonic oscillators

heenac2 said:

It doesn't actually v(t) in terms of r(t) or a(t) in terms of v(t). I'm meant to show that it's true for all simple harmonic oscillators
I know that. But, you can derive your result for part 1 by using these relationships, multiplying your starting equation by v, and integrating with respect to t. It's really simple.

Chet

## 1. How do you define position by velocity?

Position by velocity refers to the relationship between an object's position and its velocity. It is a measure of how an object's position changes over time, taking into account its direction and magnitude.

## 2. What is a constant vector?

A constant vector is a vector that remains unchanged in direction and magnitude over time. In other words, its position and velocity remain constant, hence the term "constant vector".

## 3. How can you prove that position by velocity is a constant vector?

There are several ways to prove that position by velocity is a constant vector. One way is to use the formula v = s/t, where v is the velocity, s is the displacement, and t is the time. If the velocity and displacement remain constant, then the ratio will also remain constant, indicating a constant vector. Another way is to graph the position-time and velocity-time graphs and observe if they are straight lines, which would also indicate a constant vector.

## 4. What are some real-life examples of constant vectors?

Some real-life examples of constant vectors include a car driving at a constant speed on a straight road, a person walking at a constant pace in a straight line, and a satellite orbiting the Earth at a constant speed and direction.

## 5. Why is understanding position by velocity important in science?

Understanding position by velocity is important in science because it helps us to describe and predict the motion of objects. It is a fundamental concept in physics and is used in many fields such as engineering, astronomy, and mechanics. It also allows us to understand the relationship between an object's position and its velocity, which is crucial in understanding the laws of motion and the behavior of objects in the world around us.

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