Instantaneous speed and velocity

[takando12]

Homework Statement

The instantaneous speed is always equal to the magnitude of the instantaneous velocity at that instant. Why?

Homework Equations

The Attempt at a Solution

Speed by definition is distance/time and velocity is displacement/time. I tried to reason it out. For a finite interval of time speed is greater than or equal to velocity. But When we talk about instantaneous speed or velocity, the time interval is really short.At that short an interval, the distance and displacement has got to be the same? Is this right?

 

[Qwertywerty]

Yes it is ( more or less ) .

Speed at any time is equal to magnitude of velocity . But instantaneous speed is always equal to instantaneous velocity .

Imagine a point - can you assign a direction to it ?

I hope this helps .

 

[ehild]

Homework Statement

The instantaneous speed is always equal to the magnitude of the instantaneous velocity at that instant. Why?

Homework Equations

The Attempt at a Solution

Speed by definition is distance/time and velocity is displacement/time. I tried to reason it out. For a finite interval of time speed is greater than or equal to velocity. But When we talk about instantaneous speed or velocity, the time interval is really short.At that short an interval, the distance and displacement has got to be the same? Is this right?

The displacement is a vector, the distance traveled is scalar. They are not the same.

The average speed v_av is Δs, the distance traveled by time Δt, divided by Δt. You get the instantaneous speed v if you take the limit when $Δt \rightarrow 0$ :
$$v = \lim_{Δt \rightarrow 0} \frac{Δs}{Δt}$$
The speed is scalar.

The average velocity is the displacement from A to B divided by Δt. It is a vector. The instantaneous velocity is the limit
$$\vec v = \lim_{Δt \rightarrow 0} \frac{\vec {Δr }}{Δt}$$

In the figure, the distance traveled between points A and B is equal to the length of arc AB. If the time Δt is very short the length of the displacement vector from A to B tends to the length of the arc AB. So the magnitude of the instantaneous velocity is equal to the instantaneous speed. And the velocity vector is parallel to the tangent of the curve at point A.

 

[Qwertywerty]

No - instantaneous speed is equal to instantaneous velocity , not just it's magnitude , i.e. they trace out the same path at that instant .

Speed is always the same as magnitude of velocity .

 

[HallsofIvy]

Frankly, Qwertywerty, I don't understand what you are trying to say here. First, why "more or less"? Why is this not exactly true? Second, you say
"Speed at any time is equal to magnitude of velocity . But instantaneous speed is always equal to instantaneous velocity".
I assume you meant "magnitude of instantaneous velocity" but how are those two sentences different? And what does a point not having a direction have to do with this? You seem to be confused as to what "instantaneous velocity" and "instantaneous speed" are. Just like "speed" and "velocity"
(you mean average "speed" and "velocity" don't you?) instantaneous speed is a number and instantaneous velocity is a vector. They can't be equal.

By the chain rule, $\frac{d}{dt}\sqrt{v_x^2+ v_y^2+ v_z^2}$$= \frac{2v_x\frac{dv_x}{dt}+ 2v_y\frac{dv_y}{dt}+ 2v_z\frac{dv_z}{dt}}{2\sqrt{v_x^2+ v_y^2+ v_z^2}}$$= \frac{\vec{v}}{|v|}\cdot\frac{d\vec{v}}{dt}$
The far left is the rate of change of distance- i.e. speed. The far right is the dot product of the velocity vector with a unit vector in the direction of the velocity vector- i.e. the magnitude of the velocity vector.

 

[Chestermiller]

The simple answer is that velocity is a vector, while speed is a scalar, so you can't really compare them. However, speed is equal to the magnitude of the velocity vector.

Chet

 

[THE BEAST]

i really don't understand why everyone is saying speed is equal to the magnitude of velocity vector. Are you saying it in reference to this question, in which case you are right.
Otherwise, in a generalised situation, it seems wrong to me.

 

[Chestermiller]

i really don't understand why everyone is saying speed is equal to the magnitude of velocity vector. Are you saying it in reference to this question, in which case you are right.
Otherwise, in a generalised situation, it seems wrong to me.

Sorry it seems wrong to you. Any particular reason?

Chet

 

[THE BEAST]

Sorry it seems wrong to you. Any particular reason?

Chet

Well, if a body is moving in a straight line and reverses it's direction, then in this case, won't it's speed and velocity be different?

 

[HallsofIvy]

i really don't understand why everyone is saying speed is equal to the magnitude of velocity vector. Are you saying it in reference to this question, in which case you are right.
Otherwise, in a generalised situation, it seems wrong to me.

Well, if a body is moving in a straight line and reverses it's direction, then in this case, won't it's speed and velocity be different?

Do you see the difference between these two posts? We are NOT saying that "speed' and "velocity" are the same, we are saying that speed and the magnitude of velocity are the same because, in physics, "speed" is defined as the magnitude of the velocity.

If an object is moving along the x- axis with velocity $$10\vec{i}+ 0\vec{j}$$ m/s then its speed is 10 m/s, the magnitude of the velocity vector. If it reverses its direction, so its velocity sis $$-10\vec{i}+ 0\vec{j}$$ m/s then its speed is still 10 m/s.

 

[THE BEAST]

Frankly, Qwertywerty, I don't understand what you are trying to say here. First, why "more or less"? Why is this not exactly true? Second, you say
"Speed at any time is equal to magnitude of velocity . But instantaneous speed is always equal to instantaneous velocity".
I assume you meant "magnitude of instantaneous velocity" but how are those two sentences different? And what does a point not having a direction have to do with this? You seem to be confused as to what "instantaneous velocity" and "instantaneous speed" are. Just like "speed" and "velocity"
(you mean average "speed" and "velocity" don't you?) instantaneous speed is a number and instantaneous velocity is a vector. They can't be equal.

By the chain rule, $\frac{d}{dt}\sqrt{v_x^2+ v_y^2+ v_z^2}$$= \frac{2v_x\frac{dv_x}{dt}+ 2v_y\frac{dv_y}{dt}+ 2v_z\frac{dv_z}{dt}}{2\sqrt{v_x^2+ v_y^2+ v_z^2}}$$= \frac{\vec{v}}{|v|}\cdot\frac{d\vec{v}}{dt}$
The far left is the rate of change of distance- i.e. speed. The far right is the dot product of the velocity vector with a unit vector in the direction of the velocity vector- i.e. the magnitude of the velocity vector.

is it possible to say that the interval of time is so small that the path becomes a straight line, so that the distance and displacement becomes equal and changing the direction of the body in that small instant isn't possible, so velocity becomes a scalar quantity?
I've just started reading kinematics and most textbooks haven't quite addressed this thing properly, and I'm yet to gather the courage to figure things out mathematically.
okay. wait. is magnitude of instantaneous velocity always considered with instantaneous speed ,a not just instantaneous velocity, because velocity is negative direction?

 

[Mark44]

is it possible to say that the interval of time is so small that the path becomes a straight line, so that the distance and displacement becomes equal and changing the direction of the body in that small instant isn't possible, so velocity becomes a scalar quantity?

This makes no sense. Velocity is always a vector quantity -- it can't become a scalar.

i've just started reading kinematics and most textbooks haven't quite addressed this thing properly, and I'm yet to gather the courage to figure things out mathematically.
okay. wait. is magnitude of instantaneous velocity always considered with instantaneous speed ,a not just instantaneous velocity, because velocity is negative direction?

Not sure what you're asking here, as what you have written is a bit garbled. Speed, being a magnitude, is always nonnegative, so the speed can be positive even when the velocity is negative.

 

[THE BEAST]

Do you see the difference between these two posts? We are NOT saying that "speed' and "velocity" are the same, we are saying that speed and the magnitude of velocity are the same because, in physics, "speed" is defined as the magnitude of the velocity.

If an object is moving along the x- axis with velocity $$10\vec{i}+ 0\vec{j}$$ m/s then its speed is 10 m/s, the magnitude of the velocity vector. If it reverses its direction, so its velocity sis $$-10\vec{i}+ 0\vec{j}$$ m/s then its speed is still 10 m/s.

oh! i just realized my mistake.. i have been thinking about velocity as average velocity all this time. It's quite easy to miss the difference between the two. Most problems i have been solving has been treating av velocity as velocity.

 

[takando12]

So the final answer is that if we consider really small intervals of time, the magnitude of displacement is equal to the distance traveled (or the displacement vector tends to the arc AB as pointed out by ehild) and hence the magnitude of velocity will be equal to the speed at that instant. Is this right?

 

[ehild]

the magnitude of velocity will be equal to the speed at that instant. Is this right?

It is right.