How to find total distance traveled from position function?

Steps to Solve

1. Identify the Position Function

Let's denote the position function as $s(t)$, which gives the position at time $t$. For example, suppose $s(t) = t^2 + 2t$.

2. Find the Velocity Function

To find the velocity function, take the derivative of the position function with respect to time:

$$ v(t) = \frac{ds}{dt} = s'(t) $$

Using our example, the velocity would be:

$$ v(t) = \frac{d}{dt}(t^2 + 2t) = 2t + 2 $$

3. Determine Interval of Interest

Specify the interval over which you want to calculate the total distance traveled, say $[a, b]$. You will evaluate the distance traveled in this interval.

4. Identify Points Where Velocity is Zero

Set the velocity function equal to zero to find the critical points where the object changes direction:

$$ v(t) = 0 \implies 2t + 2 = 0 $$

Solving for $t$:

$$ t = -1 $$

5. Set Up the Integral for Distance

The total distance traveled is calculated by integrating the absolute value of the velocity function. If $t = -1$ is within your interval $[a, b]$, separate the integral at that point:

$$ \text{Distance} = \int_a^{-1} |v(t)| dt + \int_{-1}^{b} |v(t)| dt $$

6. Evaluate the Integral

Calculate the definite integrals. For our example, you need to evaluate:

$$ \int |2t + 2| dt $$

Assess the sign of the velocity over $[a, -1]$ and $[-1, b]$ to determine the absolute value:

$$ \Rightarrow \text{Use the appropriate expressions based on the intervals} $$

7. Combine to Find Total Distance

Add the results of your integrals from Step 5 to find the total distance traveled.