# Why is the magnitude of velocity equal to speed

#### Helicobacter

1. solved
2. All movements in our universe are caused by the four fundamental forces. When I lift my arm, by which of these four forces is that caused by?
3. Why is the magnitude of velocity equal to speed and the magnitude of average velocity not equal to average speed?

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The total distance travelled, which is, as you said, a scalar, is the length of the trajectory of the object.

#### Helicobacter

Answers to 2 & 3 are greatly appreciated...

#### chaoseverlasting

You are converting biochemical energy into mechanical energy when you lift your arm in a gravitational field. I would say that you're using electromagnetism, as your brain uses electrical impulses to tell your arm to move, for which certain amount of energy in your body is used. Since the release of that energy would involve a chemical reaction, you could also be using weak/strong nuclear forces... but I could be way off.

The third one is easier. Average speed is total distance divided by total time, where as average velocity is total displacement by total time, and total displacement is less than or equal to total displacement. The only time it can be equal is if the object is travelling in a straight line, else displacement and distance covered will be different.

#### Helicobacter

If you have an object traveling in a circular motion with constant speed, how do you figure out its instantaneous velocity at any point? (I don't think the radius is relevant since the direction changes constantly...) Is it equal to the inst. speed?

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#### Doc Al

Mentor
If you have an object traveling in a circular motion with constant speed, how do you figure out its instantaneous velocity at any point? (I don't think the radius is relevant since the direction changes constantly...) Is it equal to the inst. speed?
The velocity is always tangent to the circle at any point. The magnitude of the velocity is the speed.

#### Helicobacter

I was just wondering since the direction of the object's motion changes after every infinitesimal small point on the circle...So, the object's velocity will be equal to its speed?

EDIT: After thinking about it: is it correct to assume that velocity and speed will only be equal in circular and straight motion?
That would imply that a circle is a vector too...

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#### Doc Al

Mentor
I was just wondering since the direction of the object's motion changes after every infinitesimal small point on the circle...
Exactly right. The velocity vector (with both magnitude and direction) continually changes direction so that it is always tangent to the circle. But its magnitude--the speed--remains constant.

EDIT: After thinking about it: is it correct to assume that velocity and speed will only be equal in circular and straight motion?
That would imply that a circle is a vector too...
I don't know what you mean. Velocity is a vector; speed is a scalar. The magnitude of the (instantaneous) velocity is the speed.

Note that the average velocity is quite different than the average speed for circular motion. Since the speed is always constant, the average speed equals the instantaneous speed. But velocity is always changing. Go back to the fundamental definition of average velocity: change in displacement over time. When an object completes a circular path, what's the change in its displacement?

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#### Helicobacter

The critical thing is that only if the tangent line to the path of motion changes, you will have fluctuating velocity.
In the case of a straight line of constant motion or my illustrated example of constant circular motion you will get constant velocity.
Are these statements correct?

#### Doc Al

Mentor
The critical thing is that only if the tangent line to the path of motion changes, you will have fluctuating velocity.
I'd state it this way: If the direction or speed changes, that means velocity is changing.
In the case of a straight line of constant motion or my illustrated example of constant circular motion you will get constant velocity.
Velocity is a vector quantity. If the direction of motion changes, like it does in uniform circular motion, then velocity is not constant.

#### Helicobacter

So would it be 0 or undefined in circular motion (with constant speed)?

#### Doc Al

Mentor
So would it be 0 or undefined in circular motion (with constant speed)?
Assuming that you are referring to my earlier question:
Go back to the fundamental definition of average velocity: change in displacement over time. When an object completes a circular path, what's the change in its displacement?
If so, then the net displacement of an object that completes a circular path is zero--since it returns to where it started. And that means the average velocity over that path is zero. (But the average speed remains non-zero.)

#### Helicobacter

I'm referring to instantaneous velocity. Assuming the speed remains constant in the circular motion, will the instantaneous velocity at any point be undefined or 0.
(Sorry for that I was vague.)

#### Doc Al

Mentor
I'm referring to instantaneous velocity. Assuming the speed remains constant in the circular motion, will the instantaneous velocity at any point be undefined or 0.
(Sorry for that I was vague.)
The instantaneous velocity of an object in uniform circular motion is perfectly well defined (it is certainly not zero!):
its magnitude equals the speed (magnitude doesn't change)
its direction depends on where it is along the path--it is always tangent to the path at any point (direction continually changes)​

#### Helicobacter

The instantaneous velocity of an object in uniform circular motion is perfectly well defined (it is certainly not zero!):
If the direction of motion changes, like it does in uniform circular motion, then velocity is not constant.
According to these statements, the inst. velocity in a constant circular motion will be defined but fluctuating.

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#### Doc Al

Mentor
According to these statements, the inst. velocity in a constant circular motion will be defined but fluctuating.
That is correct.

#### Helicobacter

There has to be a pattern. Is it a trig function?

By the way: You are very helpful, Doc Al. Thanks for your assistance!

#### Doc Al

Mentor
There has to be a pattern. Is it a trig function?
How about this. Imagine that the object is traveling in a counterclockwise circle about some origin with constant speed v. Measure the position of the object by its angle with respect to the x-axis. Then, at any point, its velocity vector (in x and y coordinates) is given by:

$$\vec{v} = -v\sin\theta \hat{x} + v\cos\theta \hat{y}$$

Does that help?