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High School Calculus Test Questions: Displacement vs. Total Distance

Hey there! 👋 Struggling with displacement vs. total distance in calculus? Don't worry, it's a common challenge! This study guide and quiz will help you ace your next test! Let's get started! 🚀
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📚 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
Click to see Answers
  1. A
  2. D
  3. C
  4. B
  5. C
  6. C
  7. A

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