# Rates of Change, particle velocity/distance?

• shocklightnin
In summary, the position of a particle is given by the equation s = 2t^3 + 36t^2 − 168t and the total distance traveled by the particle after the first seventeen seconds is 17550 meters. This is calculated by taking the difference between the position at t=17 and t=0, taking into account the direction changes of the particle.
shocklightnin

## Homework Statement

The position of a particle is given by the equation
s = 2t^3 + 36t^2 − 168t
where t is measured in seconds and s in meters.

What is the total distance traveled by the particle after the first seventeen seconds?

## Homework Equations

s(t) = 2t^3 + 36t^2 − 168t
s'(t) = 6t^2+72t-168 = v(t)

## The Attempt at a Solution

ive set t = 17 but it doesn't give me the right answer. (i get 17374). what am i doing wrong?! i know that the particle is at rest after 2 seconds (by taking derivative of this function to get the velocity function). the particle is moving forward in time interval t>2 and moving backward in time interval t<2.R

If you calculated s(17) - s(0), that gives the change in position between t = 0 and t = 17. But that isn't the total distance traveled unless the particle never reverses direction. If it changes direction you have to include the distance going to and fro too.

so calculating the direction is from 0-2 and 2-17? so it would be:
176+176+17550? (for total distance without any signs to indicate direction)
does that mean the particle went backwards 176, then forwards 176, then forwards 17550 from point 2 to 17?

shocklightnin said:
so calculating the direction is from 0-2 and 2-17? so it would be:
176+176+17550? (for total distance without any signs to indicate direction)
does that mean the particle went backwards 176, then forwards 176, then forwards 17550 from point 2 to 17?

That's the right calculation except you might re-check you calculation for ##t=17##.

## 1. What is the definition of rate of change?

The rate of change is the measure of how a quantity changes over time. It is typically denoted by the symbol "Δ" and can be calculated by dividing the change in the quantity by the change in time.

## 2. How is the rate of change related to particle velocity?

The rate of change is directly related to particle velocity. In fact, particle velocity can be defined as the rate of change of an object's position with respect to time.

## 3. What is the difference between average rate of change and instantaneous rate of change?

The average rate of change is calculated over a specific interval of time, while the instantaneous rate of change is calculated at a specific point in time. The average rate of change gives an overall picture of how the quantity changes over time, while the instantaneous rate of change gives the exact rate of change at a specific moment.

## 4. How can we use rates of change to analyze motion?

Rates of change, specifically particle velocity and distance, can be used to analyze motion by providing information about an object's speed and direction of movement. By calculating the rate of change at different points in time, we can determine if the object is speeding up, slowing down, or moving at a constant speed.

## 5. Can rates of change be negative?

Yes, rates of change can be negative. A negative rate of change indicates that the quantity is decreasing over time, while a positive rate of change indicates that the quantity is increasing over time. For example, if the rate of change of distance is negative, it means that the object is moving backwards or towards its starting point.

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