📚 Topic Summary
In calculus, displacement and total distance are related concepts but represent different quantities. Displacement refers to the change in position of an object, considering only the initial and final positions. It can be positive, negative, or zero. Total distance, on the other hand, is the entire length of the path traveled by the object, regardless of direction. It is always non-negative. To find displacement, we integrate the velocity function over a given interval. To find the total distance, we integrate the absolute value of the velocity function over the same interval. Understanding when to use each concept is vital for solving motion problems.
🧠 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Velocity |
a. The total length of the path traveled by an object. |
| 2. Displacement |
b. The rate of change of position with respect to time. |
| 3. Total Distance |
c. The integral of the absolute value of velocity over a time interval. |
| 4. Position Function |
d. The change in position of an object. |
| 5. Integral |
e. A function representing the location of an object at a given time. |
✍️ Part B: Fill in the Blanks
Fill in the blanks with the correct words.
To find the __________, integrate the velocity function $v(t)$ over the interval $[a, b]$. This gives you the net change in __________. To find the __________ __________, integrate the __________ __________ of the velocity function $|v(t)|$ over the interval $[a, b]$. This ensures you're adding up all the distances traveled, regardless of __________.
🤔 Part C: Critical Thinking
Explain, in your own words, a scenario where displacement would be zero, but the total distance traveled would be a positive value. Provide a real-world example.