High School Calculus Test Questions: Displacement vs. Total Distance

📚 Quick Study Guide

• 📏 Displacement: Displacement is the change in position of an object. It's a vector quantity, meaning it has both magnitude and direction. It can be calculated as $displacement = final\ position - initial\ position$.
• 🧭 Total Distance: Total distance is the total length of the path traveled by an object. It's a scalar quantity, meaning it only has magnitude. To find total distance, you need to consider the absolute value of the velocity function over the time interval.
• 🚀 Key Formulas:
  • Displacement: $\Delta x = \int_{t_1}^{t_2} v(t) \, dt$
  • Total Distance: $\int_{t_1}^{t_2} |v(t)| \, dt$
• ⚠️ Important Note: Displacement can be zero even if the total distance is not (e.g., an object returns to its starting point).
• 💡 Tip: Graphing the velocity function can help visualize when the object is moving in positive or negative directions, which is crucial for calculating total distance.

Practice Quiz

1. Question 1: A particle moves along the x-axis with velocity $v(t) = t^2 - 4t + 3$ for $0 \le t \le 3$. What is the displacement of the particle?
   1. -2
   2. 2
   3. 0
   4. 4
2. Question 2: Using the same velocity function as above, $v(t) = t^2 - 4t + 3$ for $0 \le t \le 3$, what is the total distance traveled by the particle?
   1. 2
   2. 4
   3. $\frac{10}{3}$
   4. $\frac{8}{3}$
3. Question 3: A car travels 10 miles east and then 5 miles west. What is the car's displacement?
   1. 15 miles east
   2. 5 miles west
   3. 5 miles east
   4. 15 miles west
4. Question 4: A particle's velocity is given by $v(t) = 2t - 4$. Find the total distance traveled from $t = 0$ to $t = 3$.
   1. 2
   2. 5
   3. 6
   4. 10
5. Question 5: An object moves with velocity $v(t) = \sin(t)$ from $t = 0$ to $t = \pi$. What is the total distance traveled?
   1. 0
   2. 1
   3. 2
   4. -2
6. Question 6: A ball is thrown upwards with an initial velocity of 20 m/s. Assuming the acceleration due to gravity is -10 m/s², what is the displacement of the ball when it returns to the point from which it was thrown?
   1. 40 m
   2. 20 m
   3. 0 m
   4. -20 m
7. Question 7: The velocity of a particle is given by $v(t) = 3t^2 - 6t$. Find the displacement of the particle between $t = 0$ and $t = 2$.
   1. -4
   2. 0
   3. 2
   4. -2