AP Physics C Mechanics- 1.1 Scalars and Vectors in One Dimension- Study Notes- New Syllabus
AP Physics C Mechanics- 1.1 Scalars and Vectors in One Dimension – Study Notes
AP Physics C Mechanics- 1.1 Scalars and Vectors in One Dimension – Study Notes – per latest Syllabus.
Key Concepts:
- Scalars and Vectors
- Vector Representation, Unit Vector Notation, and Resolution of a Vector
- Vector Operations
Scalars and Vectors
Scalar quantities are physical quantities that are described completely by magnitude (numerical value and unit) only.
Vector quantities are physical quantities that require both magnitude and direction for their complete description.
- Vectors are visually represented by arrows, where:
- The length of the arrow represents the magnitude of the vector.
- The direction of the arrow indicates the direction of the physical quantity.
Examples:
| Scalar Quantities | Vector Quantities |
|---|---|
| Distance | Displacement |
| Speed | Velocity |
| Mass | Acceleration |
| Time | Force |
| Temperature | Momentum |
Key Differences:
- Scalars combine through simple arithmetic (addition or subtraction).
- Vectors combine geometrically — their resultant depends on both magnitude and direction.
- Vector quantities obey the rules of vector addition and can be represented graphically or in component form.
- Scalars have no directional information, whereas vectors are defined by their orientation in space.
Key Idea: Scalars provide only “how much” of a quantity, while vectors provide both “how much” and “in which direction.” Understanding this distinction is essential in analyzing motion, forces, and fields in physics.
Example:
A car travels 4 km east, then 3 km north. Identify which of the following are scalar and vector quantities, and determine whether the total displacement and total distance traveled are the same or different.
▶️ Answer / Explanation
- Distance measures the total path length covered → it is a scalar quantity because it has only magnitude.
- Displacement represents the shortest straight-line path from the starting point to the ending point → it is a vector quantity because it has both magnitude and direction.
Step 2: Compare distance and displacement.
- The car travels 4 km east and then 3 km north — the total distance is the sum of both path segments (7 km).
- However, the displacement is the straight-line path from the start to the final position — it is less than 7 km and points in a direction between east and north.
Step 3: Interpretation.
- Distance depends on the total path taken and does not have direction → scalar.
- Displacement depends only on the initial and final positions and includes direction → vector.
- Thus, distance and displacement are different physical quantities even though they both describe motion.
Final Conclusion: Distance = scalar (total path length); Displacement = vector (straight-line change in position). The distance is greater than or equal to the magnitude of displacement.
Vector Representation, Unit Vector Notation, and Resolution of a Vector
Vectors are physical quantities that have both magnitude and direction. They can be represented by arrows or written in unit vector notation using standard coordinate axes.
- Each vector can be decomposed into components along the x-, y-, and z-directions.
Unit Vector Notation:
Unit vectors specify direction along each coordinate axis:
- \( \mathrm{\hat{i}} \) → x-direction
- \( \mathrm{\hat{j}} \) → y-direction
- \( \mathrm{\hat{k}} \) → z-direction
Any vector \( \mathrm{\vec{A}} \) can be expressed as a combination of these unit vectors:
\( \mathrm{\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}} \)
The magnitude of the vector is given by:
\( \mathrm{|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}} \)
Unit Vector Formula:
The unit vector in the direction of any vector \( \mathrm{\vec{A}} \) is given by:
\( \mathrm{\hat{A} = \dfrac{\vec{A}}{|\vec{A}|}} \)
- This unit vector has a magnitude of 1 and describes direction only.
Position Vector:
The position vector \( \mathrm{\vec{r}} \) locates a point in space relative to an origin.
It is expressed as:
\( \mathrm{\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}} \)
The unit vector in the direction of the position vector is:
\( \mathrm{\hat{r} = \dfrac{\vec{r}}{|\vec{r}|}} \)
Here, \( \mathrm{|\vec{r}|} \) gives the straight-line distance of the point from the origin.
Resolution of a Vector
Resolution means splitting a vector into two perpendicular components, usually along the x- and y-axes.
If a vector \( \vec{A} \) makes an angle \( \theta \) with the x-axis:
- Horizontal component: \( A_x = A \cos\theta \)
- Vertical component: \( A_y = A \sin\theta \)
- Thus, \( \vec{A} = A_x \hat{i} + A_y \hat{j} \).
The original vector can be reconstructed: \( A = \sqrt{A_x^2 + A_y^2}, \quad \tan\theta = \dfrac{A_y}{A_x} \).
Example:
A ship sails \( \mathrm{40 \, km} \) east, then \( \mathrm{30 \, km} \) north. Represent both displacements using unit vector notation and determine the resultant vector conceptually.
▶️ Answer / Explanation
Step 1: Represent the vectors.
- Eastward displacement: \( \mathrm{\vec{A} = 40 \hat{i}} \)
- Northward displacement: \( \mathrm{\vec{B} = 30 \hat{j}} \)
Step 2: Add the vectors component-wise.
\( \mathrm{\vec{R} = \vec{A} + \vec{B} = 40 \hat{i} + 30 \hat{j}} \)
Step 3: The resultant vector points diagonally northeast, showing the ship’s direct displacement from its starting point.
Step 4: The magnitude represents the straight-line distance, and the direction indicates the angle north of east.
Final Concept: The resultant vector \( \mathrm{\vec{R}} \) combines both perpendicular components and represents the net displacement in a single vector form.
Vector Operations:
Vectors can be added or subtracted either graphically (tip-to-tail method) or analytically (using components).
Addition by Vectors:
Given two vectors \( \mathrm{\vec{A}} \) and \( \mathrm{\vec{B}} \):
\( \mathrm{\vec{R} = \vec{A} + \vec{B}} \)
Component form:
\( \mathrm{\vec{R} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k}} \)
The resultant vector \( \mathrm{\vec{R}} \) represents the combined effect of \( \mathrm{\vec{A}} \) and \( \mathrm{\vec{B}} \).
Subtraction of Vectors:
Subtraction is performed by adding the negative of a vector:
\( \mathrm{\vec{R} = \vec{A} – \vec{B} = (A_x – B_x)\hat{i} + (A_y – B_y)\hat{j} + (A_z – B_z)\hat{k}} \)
Geometrically, this represents the vector from the tip of \( \mathrm{\vec{B}} \) to the tip of \( \mathrm{\vec{A}} \).
Resultant Vector:
- The resultant vector is the single vector equivalent to the combined effect of multiple vectors.
- It can be found by adding their components along each axis and then determining its magnitude and direction.
- In two dimensions, the resultant often represents the shortest path or net effect between an initial and final position.
- In a given one-dimensional coordinate system, opposite directions are denoted by opposite signs.
Example
A drone flies \(100\ \mathrm{m}\) north, then \(100\ \mathrm{m}\) northeast. Represent both displacements using unit vector notation and determine the resultant vector (magnitude and direction).
▶️ Answer / Explanation
- Northward displacement: \( \mathrm{\vec{A} = 100\,\hat{j}} \).
- Northeast displacement (45° from east): \( \mathrm{\vec{B} = 100(\cos 45^\circ\,\hat{i} + \sin 45^\circ\,\hat{j})} \).
Step 2: Compute components (numerically).
- \( \mathrm{B_x = 100\cos 45^\circ = 100\cdot \dfrac{\sqrt{2}}{2} \approx 70.71\ \mathrm{m}} \).
- \( \mathrm{B_y = 100\sin 45^\circ = 100\cdot \dfrac{\sqrt{2}}{2} \approx 70.71\ \mathrm{m}} \).
- \( \mathrm{A_x = 0,\; A_y = 100\ \mathrm{m}} \).
- \( \mathrm{R_x = A_x + B_x = 0 + 70.71 \approx 70.71\ \mathrm{m}} \).\( \mathrm{R_y = A_y + B_y = 100 + 70.71 \approx 170.71\ \mathrm{m}} \).
So in unit vector form: \( \mathrm{\vec{R} \approx 70.71\,\hat{i} + 170.71\,\hat{j}\ \mathrm{(m)}} \).
\( \mathrm{|\vec{R}| = \sqrt{R_x^2 + R_y^2} \approx \sqrt{(70.71)^2 + (170.71)^2} \approx 184.78\ \mathrm{m} \).
- Angle measured from the +x (east) axis toward the +y (north) axis: \( \mathrm{\theta = \tan^{-1}\!\left(\dfrac{R_y}{R_x}\right) \approx \tan^{-1}\!\left(\dfrac{170.71}{70.71}\right) = 67.5^\circ} \) north of east.
- Equivalently, that is \(22.5^\circ\) east of north (since \(90^\circ – 67.5^\circ = 22.5^\circ\)).