Examples of particle motion problems with derivatives AP Calculus

📚 Quick Study Guide

• 📏 Position: The position of a particle at time $t$ is given by $s(t)$.
• 🚀 Velocity: Velocity is the rate of change of position, so $v(t) = s'(t) = \frac{ds}{dt}$.
• 💨 Acceleration: Acceleration is the rate of change of velocity, so $a(t) = v'(t) = s''(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2}$.
• ➕ Speed: Speed is the absolute value of velocity, i.e., $|v(t)|$.
• ➡️ Moving Right: The particle moves to the right when $v(t) > 0$.
• ⬅️ Moving Left: The particle moves to the left when $v(t) < 0$.
• 🛑 At Rest: The particle is at rest when $v(t) = 0$.
• 🔄 Changing Direction: The particle changes direction when $v(t)$ changes sign.
• 🛤️ Displacement: Displacement from time $t_1$ to $t_2$ is given by $\int_{t_1}^{t_2} v(t) dt = s(t_2) - s(t_1)$.
• 💪 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. $e^5$