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I've been struggling to understand the differences between average velocity and average speed in 1 dimension; by doing some exercises I think I have figured it out, but i want to check if I'm correct.

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This describes the movement of a particle through 4 positions: $P_1, P_2, P_3, P_4$. Each position has a respective coordinate $x_1, x_2, x_3, x_4$, and a respective moment in time $t_1, t_2, t_3, t_4$ in which the particle is located in that position.

Let's talk about the movement from $P_1$ to $P_3$:

The x component of average velocity is $\frac{x_3 - x_1}{t_3 - t_1}$. However, average speed seems to be $\frac{|x_2-x_1|+|x_3-x_2|}{|t_2-t_1|+|t_3-t_2|}$. I have some questions about the latter; As you may have seen, I used absolute value signs. Regarding the norm of the vectors (which describes the distance between the points) it is because I bumped into an exercise in which average speed, initial and final coordinates, and initial time were given, and I had to solve for final time. the final position was a negative number, and initial was positive, so I ended up getting a negative distance, which resulted in a negative final time, something which can not happen. So I concluded that average speed is about the movements in absolute value, which is reinforced by the fact it's an scalar. I'm sure putting absolute values to find the distances between points is correct, but I could be wrong, so ¿is this correct?. The ones I'm not so sure about are the absolute value signs in time, because up to this point I have not done an exercise with negative time involved, but I don't know if they exist, which if they didn't, would make abs. val. signs unnecesary ¿do you know if I have to put them? ¿will I find excercises with negative time involved?

About the movement from $P_1$ to $P_2$.

It seems x component of average velocity is $\frac{x_2 - x_1}{t_2 - t_1}$, and average speed is given by $\frac{|x_2 - x_1|}{|t_2 - t_1|}$ Is this correct?

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    $\begingroup$ I'll keep saying it forever: the desire of the English to be special will keep confusing people. Velocity is the only vector whose modulus has a special name. Other languages do not make such distinction. Other vectors do not cause those problems: force can be $F$ or $\vec{F}$ depending on the context. So this is the same. $\endgroup$
    – FGSUZ
    Nov 23 '18 at 20:10
  • $\begingroup$ The easiest way to understand the difference is to think about 1-dimensional motion along a circle rather along a straight line. A particle moving in a circle with constant speed has zero average velocity. $\endgroup$
    – G. Smith
    Nov 23 '18 at 20:10
  • $\begingroup$ @FGSUZ I assume you're joking about having the two words "speed" and "velocity" being anything to do with a desire to be special! The mean speed is not necessarily the same as the modulus of the mean velocity, as G. Smith illustrates. I'd also defend the usefulness of having separate terms "displacement" and "distance". Is English the only language to have these? $\endgroup$
    – Philip Wood
    Nov 23 '18 at 22:33
  • $\begingroup$ @PhilipWood of course it wasn't serious, hope I didn't offend you. But yep, this is an issue that keeps rising every season. I wanted to point out that everybody understands that force isn't the same as it modulus. Yet we all can use the single word "force" with both meanings and we don't get confused. So I don't se any utility on the speed/velocity distinction, rather, it confuses people over and over again. $\endgroup$
    – FGSUZ
    Nov 23 '18 at 23:07
  • $\begingroup$ No offence! Enjoyed the joke. But what about $distance$ and $displacement$? A useful distinction? $\endgroup$
    – Philip Wood
    Nov 23 '18 at 23:40
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It seems x component of average velocity is x2−x1t2−t1, and average speed is given by |x2−x1||t2−t1| ¿is this correct?

Yes, this is basically the definition. Those vertical bars should (in this case) not be thought of as "absolute value", but "magnitude". The magnitude of a velocity that is "five to the left" is five. Yes, that is the absolute value in that case, but it isn't always. For instance, the magnitude of the velocity "5 to the left and 2 up" is 5.4 - do you see why?

All of this is made easier if you always consider the motion/measurement/whatever along each axis separately. So in that case instead of saying the velocity is 5 to the left and 2 up" you would write (-5, 2), and then your speed is (5, 2). And then, hey presto, you're back to the magnitude being simply the absolute value of the numbers again. And this is why we write |v| to mean "the magnitude of the vector", because ultimately its the same.

I wish the math quoted properly...

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  • $\begingroup$ Thanks! But now I'm only left with just one doubt: The absolute value signs in the denominator of $\frac{|x_2-x_1|+|x_3-x_2|}{|t_2-t_1|+|t_3-t_2|}$, these are: $|t_2-t_1|+|t_3-t_2|$ are necessary? I ask this because I haven't bumped into an excercise with negative time, so if time is never negative, absolute value signs aren't necessary, but if time can be negative, then they might be. $\endgroup$
    – Daniel Bonilla Jaramillo
    Nov 23 '18 at 20:21
  • $\begingroup$ Time is a dimension too, so it can definitely be negative. Yes, time here in the "real world:" always goes one way, but you can still think of things that happened "five seconds ago". $\endgroup$
    – Maury Markowitz
    Nov 23 '18 at 20:28

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