Real-World Examples of Vector-Valued Functions in Physics

📚 Quick Study Guide

• 📏 Vector-valued functions describe motion in space as a function of time, $t$. Generally written as $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$.
• 🚗 Velocity is the derivative of the position vector: $\mathbf{v}(t) = \mathbf{r}'(t)$. Speed is the magnitude of velocity: $|\mathbf{v}(t)|$.
• 🚀 Acceleration is the derivative of the velocity vector: $\mathbf{a}(t) = \mathbf{v}'(t) = \mathbf{r}''(t)$.
• 💡 Projectile motion assumes constant gravitational acceleration, typically $\mathbf{g} = \langle 0, -g \rangle$ (in 2D) or $\mathbf{g} = \langle 0, 0, -g \rangle$ (in 3D), where $g \approx 9.8 \text{ m/s}^2$.
• 🍎 Newton's Second Law: $\mathbf{F}(t) = m\mathbf{a}(t)$, where $\mathbf{F}$ is the net force, $m$ is mass, and $\mathbf{a}$ is acceleration.
• 🌌 Work done by a force $\mathbf{F}$ along a path $\mathbf{r}(t)$ from $a$ to $b$ is given by the line integral: $W = \int_a^b \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) dt$.
• 🌀 Angular momentum $\mathbf{L}$ of a particle with position $\mathbf{r}$ and momentum $\mathbf{p} = m\mathbf{v}$ is given by $\mathbf{L} = \mathbf{r} \times \mathbf{p}$.

🧪 Practice Quiz

1. What does the derivative of a position vector-valued function represent?
   1. A) Acceleration
   2. B) Velocity
   3. C) Speed
   4. D) Jerk
2. A projectile is launched with initial velocity $\mathbf{v}_0 = \langle 10, 20 \rangle$ m/s. Ignoring air resistance, what is its acceleration?
   1. A) $\mathbf{a}(t) = \langle 0, 0 \rangle$
   2. B) $\mathbf{a}(t) = \langle 0, -9.8 \rangle$
   3. C) $\mathbf{a}(t) = \langle -9.8, 0 \rangle$
   4. D) $\mathbf{a}(t) = \langle 10, 0 \rangle$
3. Given a force $\mathbf{F}(t) = \langle t, t^2 \rangle$ and a displacement $\mathbf{r}(t) = \langle t^2, t \rangle$ for $0 \le t \le 1$, what is the work done?
   1. A) 1
   2. B) 2
   3. C) 0.5
   4. D) 1.5
4. If $\mathbf{r}(t) = \langle \cos(t), \sin(t) \rangle$, what is the speed of the particle?
   1. A) $\sin(t) + \cos(t)$
   2. B) 1
   3. C) $\sqrt{2}$
   4. D) $t$
5. What physical quantity is calculated by integrating the dot product of force and the derivative of the position vector with respect to time, along a path?
   1. A) Kinetic Energy
   2. B) Potential Energy
   3. C) Work Done
   4. D) Power
6. A particle moves along the path $\mathbf{r}(t) = \langle t, t^2, t^3 \rangle$. What is its velocity at $t=1$?
   1. A) $\langle 1, 1, 1 \rangle$
   2. B) $\langle 1, 2, 3 \rangle$
   3. C) $\langle 0, 2, 6 \rangle$
   4. D) $\langle 1, 4, 9 \rangle$
7. According to Newton's Second Law, which vector is directly proportional to the net force acting on an object?
   1. A) Velocity
   2. B) Position
   3. C) Momentum
   4. D) Acceleration