# How to Calculate Average Speed Using Parametric Equations

• keemosabi
In summary, the conversation discusses finding the average speed of a particle moving along a path represented by parametric equations. The formula for this is sqrt((x(b)-x(a))^2+(y(b)-y(a))^2)/(b-a), where a is the initial time and b is the final time. The conversation also considers the possibility of using the average value of the absolute value of the velocity graph to determine the speed. It is also mentioned that the speed of the particle can be determined by looking at whether the quantity sqrt((dx/dt)^2+(dy/dt)^2) is increasing or decreasing.
keemosabi

## Homework Statement

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?

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.

w3390 said:
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.

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

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

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?

Dick said:
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?

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

keemosabi said:
Couldn't you also do the average value of the absolute value of the velocity graph?

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.

Dick said:
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.

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?

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

## What is a parametric equation?

A parametric equation is a mathematical expression that describes the relationship between two or more variables. In the context of speed, a parametric equation can be used to describe the relationship between distance, time, and velocity.

## How is speed calculated using parametric equations?

In parametric equations, speed can be calculated by taking the derivative of the position equation with respect to time. This will give the instantaneous velocity at any given point in time.

## What are the advantages of using parametric equations to calculate speed?

Parametric equations allow for a more accurate and precise calculation of speed compared to using a single formula, as they take into account the change in velocity over time. They also allow for more complex and dynamic relationships between variables to be described.

## Can parametric equations be used for non-linear motion?

Yes, parametric equations can be used to describe non-linear motion, such as circular or projectile motion, by incorporating trigonometric functions into the equations.

## Are there any limitations to using parametric equations for calculating speed?

One limitation of using parametric equations is that they require more complex calculations compared to using a single formula. They also assume that the motion is continuous and smooth, which may not always be the case in real-world scenarios.

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