How to find total distance traveled from position function?
Understand the Problem
The question is asking for a method to calculate the total distance traveled based on a position function. This typically involves integrating the absolute value of the velocity function derived from the position function over a specified interval.
Answer
Total distance = $ \int_a^{-1} |v(t)| dt + \int_{-1}^{b} |v(t)| dt $
Answer for screen readers
The total distance traveled is given by:
$$ \int_a^{-1} |v(t)| dt + \int_{-1}^{b} |v(t)| dt $$
Steps to Solve
- Identify the Position Function
Let's denote the position function as $s(t)$, which gives the position at time $t$. For example, suppose $s(t) = t^2 + 2t$.
- 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:
$$ v(t) = \frac{d}{dt}(t^2 + 2t) = 2t + 2 $$
- 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.
- 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 $t$:
$$ t = -1 $$
- 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:
$$ \text{Distance} = \int_a^{-1} |v(t)| dt + \int_{-1}^{b} |v(t)| dt $$
- 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:
$$ \Rightarrow \text{Use the appropriate expressions based on the intervals} $$
- Combine to Find Total Distance
Add the results of your integrals from Step 5 to find the total distance traveled.
The total distance traveled is given by:
$$ \int_a^{-1} |v(t)| dt + \int_{-1}^{b} |v(t)| dt $$
More Information
Calculating total distance using the absolute value of velocity is important because it accounts for changes in direction. This method is applicable for any position function and can be used across a variety of problems involving motion.
Tips
- Forgetting to take the absolute value of the velocity when calculating total distance.
- Neglecting to check if critical points fall within the interval of interest.
- Miscalculating the definite integrals when changing limits due to sign changes in the velocity function.