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.
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?