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# Parametric Equation Speed

1. ### keemosabi

109
1. The problem statement, all variables and given/known data
Can someone please tell me how to get the average speed of a particle moving along a path represented by parametric equations? Is it $\frac{1}{b-a}\int_{a}^{b}\sqrt{\frac{dx }{d t}^2 + \frac{d y}{d t}^2}$

Isn't this the arc length formula?

2. ### w3390

344
This is the arc length formula. The average value formula is Favg=(1/b-a)INT[f(x)dx]. It seems you combined two formulas.

3. ### keemosabi

109
But if I wanted the speed of a particle moving with a parametric graph, woldn't everything under the radical be my speed function?

4. ### w3390

344
Actually, you may be right. I think that might actually work.

5. ### Dick

25,833
No, no, no. The average speed is displacement over time. It has nothing to do with arc length. It's sqrt((x(b)-x(a))^2+(y(b)-y(a))^2)/(b-a) where a is the intiial time and b is the final time. Right?

6. ### keemosabi

109
Couldn't you also do the average value of the absolute value of the velocity graph?

7. ### Dick

25,833
Yes, you could. In which case that would be correct. Distance travelled/time could also be considered an average speed. I was only thinking of the displacement/time definition.

8. ### keemosabi

109
Alirght, thank you for the help.

Also, is there any way to determine if a particle traveling on a parametric path is increasing in speed? I know I can determine if the x and y are accelerating, but I can I determine if the particle itself is increasing?

What if it was accelearating in the x direction but decelerating in the y? Would the particle's speed be increasing or decreasing?

9. ### Dick

25,833
The 'speed' is sqrt((dx/dt)^2+(dy/dt)^2), isn't it? Just look at whether that quantity is increasing or decreasing.