#AP Physics C: Mechanics - Scalars and Vectors 🚀

Hey there! Let's make sure you're totally solid on scalars and vectors—these are the building blocks of everything we'll do in mechanics. Think of this as your cheat sheet for tonight!

#Scalars vs. Vectors: The Basics

• Scalars: These are quantities that have only magnitude (size). Think of them as just a number with a unit. Examples include:

  • Mass (e.g., 5 kg)
  • Time (e.g., 10 s)
  • Temperature (e.g., 25°C)
  • Energy (e.g., 100 J)
  • Work (e.g., 50 Nm)
  • Electric charge (e.g., 1.6e-19 C)
  • 📏 Think: Just a number!

• Vectors: These have both magnitude and direction. They're like arrows pointing somewhere with a certain length. Examples include:

  • Displacement (e.g., 5 m, north)
  • Velocity (e.g., 10 m/s, east)
  • Acceleration (e.g., 9.8 m/s², down)
  • Force (e.g., 20 N, at 30°)
  • Momentum (e.g., 10 kg m/s, west)
  • Electric field (e.g., 100 N/C, right)
  • 🧭 Think: Number + Direction!

• Scalars can be added, subtracted, multiplied, and divided just like regular numbers.
• Vectors need special rules for addition and subtraction because of their direction. This is super important for analyzing motion and forces!

• Scalar: Size only, Simple math
• Vector: Velocity has direction, Very specific math

#Vector Representation

#Distance vs. Displacement & Speed vs. Velocity

• Distance (scalar): How far something has traveled in total. A runner's distance is 5 km.
• Displacement (vector): How far something is from its starting point, with a direction. A car's displacement is 10 km east.
• Speed (scalar): How fast something is moving. A car's speed is 60 mph.
• Velocity (vector): How fast something is moving in a specific direction. A car's velocity is 60 mph, north.

• We write vectors with an arrow above the symbol: $\vec{v}$ (velocity), $\vec{a}$ (acceleration).
• In 1D, we often drop the arrow and use +/- signs to indicate direction.
• Example: $\vec{v} = \vec{v}_{0} + \vec{a}t$ (vector equation) or $v_x = v_{x0} + a_xt$ (1D component equation)

• In 1D problems, you can often skip the full vector notation and just use + and - signs to show direction. This will save you time!

#Vector Notation: Breaking It Down

#Unit Vector Notation

• We use unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$ to represent the x, y, and z directions, respectively.
• A vector $\vec{r}$ can be written as: $\vec{r} = A\hat{i} + B\hat{j} + C\hat{k}$, where A, B, and C are the scalar components in each direction.
• The magnitude of a unit vector is always 1. They are perpendicular to each other.
• Example: If $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$, then $a_x = 2$, $a_y = -3$, and $a_z = 4$.

#Magnitude and Direction

• Vectors can also be expressed by stating their magnitude and direction, e.g., "a force of 50 N at an angle of 30° above the horizontal."

• $\hat{i}$ points to the right (x-axis), $\hat{j}$ points up (y-axis), and $\hat{k}$ points out (z-axis). They're like your 3D GPS coordinates!

• Unit vectors make vector algebra easier and allow for quick conversion between different notations. They help you keep track of each component separately.

#Final Exam Focus 🎯

#High-Priority Topics

• Distinguishing between scalars and vectors: Know the difference and be able to give examples of each.
• Vector addition and subtraction: Master graphical and component methods.
• Unit vector notation: Be comfortable using $\hat{i}$, $\hat{j}$, and $\hat{k}$.
• Applying vectors to kinematic equations: Understand how direction affects motion.

#Common Question Types

• Multiple Choice: Identifying scalar vs. vector quantities, simple vector addition/subtraction.
• Free Response: Vector analysis in 2D motion, force problems involving vector components.

• Time Management: Don't spend too long on any one question. If you're stuck, move on and come back later.
• Common Pitfalls: Pay close attention to units and directions. Double-check your calculations.
• Strategies: Draw diagrams for vector problems. Break vectors into components when necessary.

Practice Question

#Practice Questions

#Multiple Choice

1. Which of the following is a scalar quantity? (A) Velocity (B) Acceleration (C) Time (D) Force

2. A car travels 20 km east and then 30 km north. What is the magnitude of the car's displacement? (A) 10 km (B) 50 km (C) 36 km (D) 100 km

3. A vector is given by $\vec{A} = 3\hat{i} - 4\hat{j}$. What is the magnitude of vector $\vec{A}$? (A) 1 (B) 5 (C) 7 (D) 25

#Free Response

A small drone flies with a velocity of $\vec{v_1} = 5\hat{i} + 3\hat{j}$ m/s. After 10 seconds, its velocity is $\vec{v_2} = 10\hat{i} - 2\hat{j}$ m/s.

(a) Determine the change in the drone's velocity vector $\Delta \vec{v}$. (2 points)

(b) Calculate the average acceleration vector $\vec{a}_{avg}$ of the drone during this 10-second interval. (3 points)

(c) Find the magnitude of the average acceleration. (2 points)

(d) If the drone started at the origin, what is the displacement vector after 10 seconds? (5 points)

#Answer Key

Multiple Choice

1. (C)
2. (C) Using Pythagorean theorem: $\sqrt{20^2 + 30^2} = \sqrt{1300} \approx 36 km$
3. (B) Magnitude is $\sqrt{3^2 + (-4)^2} = 5$

Free Response

(a) $\Delta \vec{v} = \vec{v_2} - \vec{v_1} = (10\hat{i} - 2\hat{j}) - (5\hat{i} + 3\hat{j}) = 5\hat{i} - 5\hat{j}$ (2 points: 1 for correct subtraction, 1 for correct components)

(b) $\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t} = \frac{5\hat{i} - 5\hat{j}}{10} = 0.5\hat{i} - 0.5\hat{j} \text{ m/s}^2$ (3 points: 1 for correct formula, 1 for correct substitution, 1 for correct answer)

(c) $|\vec{a}_{avg}| = \sqrt{(0.5)^2 + (-0.5)^2} = \sqrt{0.5} \approx 0.707 \text{ m/s}^2$ (2 points: 1 for correct magnitude formula, 1 for correct answer)

(d) The displacement can be found using the average velocity and the time. The average velocity is $\vec{v}_{avg} = \frac{\vec{v_1} + \vec{v_2}}{2} = \frac{15\hat{i} + \hat{j}}{2} = 7.5\hat{i} + 0.5\hat{j}$. The displacement is $\Delta \vec{r} = \vec{v}_{avg} * \Delta t = (7.5\hat{i} + 0.5\hat{j}) * 10 = 75\hat{i} + 5\hat{j} \text{ m}$ (5 points: 1 for correct average velocity, 2 for correct displacement formula, 2 for correct answer)