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gonzales.susan58 2d ago โ€ข 0 views

Examples of particle motion problems with derivatives AP Calculus

Hey there! ๐Ÿ‘‹ Trying to wrap your head around particle motion problems in AP Calc? Derivatives can be tricky, but with a solid understanding and some practice, you'll be solving these problems like a pro! Let's get started with a quick review, then test your knowledge with a quiz. Good luck!๐Ÿ€
๐Ÿงฎ Mathematics

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janet_delgado Dec 27, 2025

๐Ÿ“š Quick Study Guide

  • ๐Ÿ“ Position: The position of a particle at time $t$ is given by $s(t)$.
  • ๐Ÿš€ Velocity: Velocity is the rate of change of position, so $v(t) = s'(t) = \frac{ds}{dt}$.
  • ๐Ÿ’จ Acceleration: Acceleration is the rate of change of velocity, so $a(t) = v'(t) = s''(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2}$.
  • โž• Speed: Speed is the absolute value of velocity, i.e., $|v(t)|$.
  • โžก๏ธ Moving Right: The particle moves to the right when $v(t) > 0$.
  • โฌ…๏ธ Moving Left: The particle moves to the left when $v(t) < 0$.
  • ๐Ÿ›‘ At Rest: The particle is at rest when $v(t) = 0$.
  • ๐Ÿ”„ Changing Direction: The particle changes direction when $v(t)$ changes sign.
  • ๐Ÿ›ค๏ธ Displacement: Displacement from time $t_1$ to $t_2$ is given by $\int_{t_1}^{t_2} v(t) dt = s(t_2) - s(t_1)$.
  • ๐Ÿ’ช Total Distance: Total distance traveled from time $t_1$ to $t_2$ is given by $\int_{t_1}^{t_2} |v(t)| dt$.

Practice Quiz

  1. Question 1: The position of a particle moving along the x-axis is given by $s(t) = t^3 - 6t^2 + 9t + 1$. At what time(s) is the particle at rest?
    1. $t = 1$ and $t = 3$
    2. $t = 0$
    3. $t = 2$
    4. $t = -1$ and $t = -3$
  2. Question 2: A particle moves along the x-axis with velocity $v(t) = t^2 - 4t + 3$. During what time interval(s) is the particle moving to the left?
    1. $(1, 3)$
    2. $(-\infty, 1)$ and $(3, \infty)$
    3. $(2, \infty)$
    4. $(-\infty, 2)$
  3. Question 3: The velocity of a particle is given by $v(t) = 2t - 6$. If the position of the particle at $t = 0$ is $s(0) = 5$, what is the position of the particle at $t = 3$?
    1. $s(3) = -4$
    2. $s(3) = 5$
    3. $s(3) = -9$
    4. $s(3) = 0$
  4. Question 4: The acceleration of a particle is given by $a(t) = 6t$. If the velocity at $t = 0$ is $v(0) = -5$, and the position at $t = 0$ is $s(0) = 10$, find the position function $s(t)$.
    1. $s(t) = t^3 - 5t + 10$
    2. $s(t) = 3t^2 - 5t + 10$
    3. $s(t) = t^3 + 10$
    4. $s(t) = t^3 - 5t$
  5. Question 5: A particle's velocity is given by $v(t) = \cos(t)$. What is the total distance traveled by the particle from $t = 0$ to $t = \pi$?
    1. $0$
    2. $1$
    3. $2$
    4. $\pi$
  6. Question 6: The position of a particle is given by $s(t) = t^2 - 4t$. At what time does the particle change direction?
    1. $t = 2$
    2. $t = 0$
    3. $t = 4$
    4. $t = -2$
  7. Question 7: A particle moves along the x-axis such that its velocity is given by $v(t) = e^{\sin(t)}\cos(t)$ for $0 \leq t \leq 5$. What is the displacement of the particle from $t=0$ to $t=5$?
    1. $e^{\sin(5)}-1$
    2. $e^{\sin(5)}$
    3. $e^5-1$
    4. $e^5$
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