Comprehensive Bullet-Point Notes — Chapter 3 : Motion in a Plane

Scalars and Vectors

• What are quantities?

  • Scalar: Just has a "how much" (magnitude), no direction. You describe it with a single number and a unit.

    • Examples: distance, weight, heat, time, size, how thick something is.

  • Vector: Has both a "how much" (magnitude) and a "which way" (direction). It follows special rules for adding them (like the triangle or parallelogram rule).

    • Examples: how far you moved, speed, how fast speed changes, push/pull, how hard something is moving, electric effect.

• Working with scalars

  • Adding, subtracting, multiplying, and dividing scalars works like regular math.

  • You can only add or subtract scalars if they have the same units. You can multiply or divide them even if units are different (like how density is weight divided by size).

• How to write vectors

  • In print: big bold letter $\mathbf{A}$. By hand: letter with an arrow on top $\vec{A}$.

  • Its "how much" (magnitude) is written as $||\mathbf{A}|| = A$.

Position & Displacement Vectors

• Pick a starting point $O$. The position of a point $P$ at a certain time $t$ is shown by the arrow $\vec{r} = \overrightarrow{OP}$.

• A new position $P'$ at a later time $t'$ is $\vec{r'} = \overrightarrow{OP'}$.

• The change in position from $P$ to $P'$ (displacement) is:
  
  $\Delta \vec{r} = \vec{r'} - \vec{r} = \overrightarrow{PP'}$.

• It doesn't matter what path you take between $P$ and $P'$; the displacement $\Delta \vec{r}$ will always be the same.

• The size of the displacement is always less than or equal to the actual path length you traveled.

Equality of Vectors

• Two vectors $\mathbf{A}$ and $\mathbf{B}$ are equal if they have the same "how much" (magnitude) AND point in the exact same direction.

• Free vectors: Their exact spot doesn't matter; you can move them anywhere parallel to themselves without changing them.

• Localised vectors: Their exact line of action matters (like forces pushing on a specific part of an object).

Multiplication by Real Numbers

• If $\lambda$ is a positive number (\lambda > 0): The size of $\lambda \mathbf{A}$ is $\lambda$ times the size of $||\mathbf{A}||$ ($||\lambda \mathbf{A}|| = \lambda ||\mathbf{A}|||$), and the direction stays the same.

• If $\lambda$ is a negative number (\lambda < 0): The size is $|\lambda|$ times the size of $||\mathbf{A}||$ ($||\lambda|| ||\mathbf{A}||=$), and the direction flips the other way.

• What happens to units? Units also multiply (for example, $\text{(meters/second)} \times \text{seconds} = \text{meters}$).

Graphical Vector Operations

Adding Vectors (Head–to-Tail / Triangle Method)

• To add $\mathbf{A}$ and $\mathbf{B}$, put the tail (start) of vector $\mathbf{B}$ at the head (end) of vector $\mathbf{A}$. The resulting vector $\mathbf{R} = \mathbf{A} + \mathbf{B}$ goes from the tail of $\mathbf{A}$ to the head of $\mathbf{B}$.

• Rules for Addition:

  • Order doesn't matter: $\mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}$.

  • Grouping doesn't matter: $(\mathbf{A}+\mathbf{B})+\mathbf{C}=\mathbf{A}+(\mathbf{B}+\mathbf{C})$.

• Zero/Null vector $\mathbf{0}$:

  • If you add a vector to its negative (same size, opposite direction), you get the zero vector: $\mathbf{A}+(-\mathbf{A})=\mathbf{0}$. Its size is $0$ ($||\mathbf{0}||=0$), and it has no defined direction.

• Subtracting Vectors:

  • Subtracting $\mathbf{B}$ from $\mathbf{A}$ is the same as adding $\mathbf{A}$ to the negative of $\mathbf{B}$: $\mathbf{A}-\mathbf{B}=\mathbf{A}+(-\mathbf{B})$.

Parallelogram Method

• Put the tails of $\mathbf{A}$ and $\mathbf{B}$ at the same starting point. The diagonal of the parallelogram they form will be their sum $\mathbf{A}+\mathbf{B}$.

Breaking Down Vectors (Resolution)

• Any vector $\mathbf{A}$ in a flat space (a plane) can be broken down into parts using two basis vectors $\mathbf{a}$ and $\mathbf{b}$ that don't point in the same line: $\mathbf{A}=\lambda \mathbf{a}+\mu \mathbf{b}$.

Unit Vectors

• These are special vectors of length 1 that point along the x, y, and z axes: $\hat{i}, \hat{j}, \hat{k}$. Their magnitudes are 1: $||\hat{i}||=||\hat{j}||=||\hat{k}||=1$.

• Breaking down in 2-D:
  The vector $\mathbf{A}$ can be written as its x-part times $\hat{i}$ plus its y-part times $\hat{j}$:
  $\mathbf{A}=A<em>x \hat{i}+A</em>y \hat{j}$
  Where $A<em>x$ is the part along the x-axis ($A</em>x=A \cos\theta$) and $A<em>y$ is the part along the y-axis ($A</em>y=A \sin\theta$). $\theta$ is the angle the vector makes with the x-axis.

• In 3-D:
  In 3D, it's $\mathbf{A}=A<em>x \hat{i}+A</em>y \hat{j}+A<em>z \hat{k}$. Its total size (magnitude) is $A=\sqrt{A</em>x^2+A<em>y^2+A</em>z^2}$.

• Direction cosines: These are the cosines of angles the vector makes with the x, y, and z axes ($\alpha,\beta,\gamma$). They follow the rule: $\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$.

Adding Vectors Analytically (Using Components)

• If you have two vectors $\mathbf{A}=A<em>x \hat{i}+A</em>y \hat{j}$ and $\mathbf{B}=B<em>x \hat{i}+B</em>y \hat{j}$, their sum $\mathbf{R}=\mathbf{A}+\mathbf{B}$ is found by adding their x-parts and their y-parts separately:
  $\mathbf{R}=\mathbf{A}+\mathbf{B}=(A<em>x+B</em>x)\hat{i}+(A<em>y+B</em>y)\hat{j}$.

• This method works for 3-D vectors too.

• Law of Cosines & Sines: These laws help find the size (magnitude) and angle of the resulting vector when you add two vectors $A$ and $B$ that are at an angle $\theta$ to each other:
  $R^2=A^2+B^2+2AB\cos\theta$
  $\frac{R}{\sin\theta}=\frac{B}{\sin\beta}=\frac{A}{\sin\alpha}$.

Motion in a Flat Space (Kinematics in a Plane)

Position & Displacement

• Position is shown as $\vec{r}(t)=x(t)\hat{i}+y(t)\hat{j}$, meaning it changes with time.

• Displacement (change in position) is $\Delta \vec{r}=\vec{r}(t+\Delta t)-\vec{r}(t)$.

Velocity

• Average Velocity: \vec{v}__{avg}=\frac{\Delta \vec{r}}{\Delta t} (its direction is the same as the direction of $\Delta \vec{r}$).

• Instantaneous Velocity: This is the velocity at a specific moment. It's the rate of change of position:
  $\vec{v}=\lim<em>_{\Delta t\to 0}\frac{\Delta \vec{r}}{\Delta t}=\frac{d\vec{r}}{dt}=v</em>x\hat{i}+v<em>y\hat{j}$ Here, $v</em>x$ is how fast it's moving in the x-direction ($v<em>x=\frac{dx}{dt}$), and $v</em>y$ is how fast it's moving in the y-direction ($v_y=\frac{dy}{dt}$).

• The speed (magnitude of velocity) is $v=\sqrt{v<em>x^2+v</em>y^2}$. The velocity vector always points along the path the object is taking.

Acceleration

• Average Acceleration: \vec{a}__{avg}=\frac{\Delta \vec{v}}{\Delta t}.

• Instantaneous Acceleration: This is the acceleration at a specific moment. It's the rate of change of velocity:
  $\vec{a}=\frac{d\vec{v}}{dt}=a<em>x\hat{i}+a</em>y\hat{j}$
  Here, $a<em>x$ is how fast the x-velocity changes ($a</em>x=\frac{dv<em>x}{dt}$), and $a</em>y$ is how fast the y-velocity changes ($a<em>y=\frac{dv</em>y}{dt}$).

• In general, the direction of velocity $\vec{v}$ and acceleration $\vec{a}$ don't have to be the same. The angle between them can change from $0^\circ$ to $180^\circ$.

Motion with Constant Acceleration in 2-D

• If the acceleration $\vec{a}$ is always the same, and you know the initial velocity $\vec{v}<em>0$ and initial position $\vec{r}</em>0$:

  • Velocity: $\vec{v}=\vec{v}<em>0+\vec{a}t$ (For parts: $v</em>x=v<em>{0x}+a</em>xt$ and $v<em>y=v</em>{0y}+a_yt$).

  • Position: $\vec{r}=\vec{r}<em>0+\vec{v}</em>0 t+\tfrac12 \vec{a} t^2$ (The parts work the same way).

• This kind of motion is like having two separate movements happening at the same time: one in the x-direction and one in the y-direction, both with steady acceleration.

Projectile Motion (Ignoring Air Resistance)

• Starting: An object is launched with an initial speed $v<em>0$ at an angle $\theta</em>0$ above the ground.

• Acceleration: The only acceleration is due to gravity, pointing straight down: $a<em>x=0$ (no horizontal acceleration), and $a</em>y=-g$ (vertical acceleration is negative because gravity pulls down).

Equations for Position and Velocity over Time

• Position at time $t$:
  $x=(v<em>0\cos\theta</em>0)t$
  $y=(v<em>0\sin\theta</em>0)t-\tfrac12 g t^2$

• Velocity at time $t$:
  $v<em>x=v</em>0\cos\theta<em>0$ (This horizontal velocity stays constant) $v</em>y=v<em>0\sin\theta</em>0-g t$ (Vertical velocity changes due to gravity)

Path of the Object (Trajectory)

• If you remove time $t$ from the position equations, you get the shape of the path, which is a parabola:
  $y= x\tan\theta<em>0- \frac{g x^2}{2 v</em>0^2 \cos^2\theta_0}$

Important Results

• Time to reach highest point: $t<em>m=\frac{v</em>0\sin\theta_0}{g}$.

• Total time in the air (flight time): $T<em>f=\frac{2v</em>0\sin\theta<em>0}{g}=2t</em>m$.

• Maximum height reached: $h<em>{m}=\frac{v</em>0^{2}\sin^{2}\theta_0}{2g}$.

• Horizontal distance traveled (range): $R=\frac{v<em>0^{2}\sin 2\theta</em>0}{g}$. This distance is largest when launched at $45^{\circ}$ ($\theta<em>0=45^{\circ}$), giving a maximum range of $R</em>{max}=\frac{v_0^{2}}{g}$.

• Complementary angles: Launching at angles like $30^\circ$ and $60^\circ$ (which add up to $90^\circ$) will give the same horizontal range.

Uniform Circular Motion (UCM)

• An object moves at a steady speed $v$ around a circle with radius $R$.

Angular Measurements

• Angular distance: How much angle it has covered ($\Delta\theta$).

• Angular speed: How fast the angle is changing ($\omega=\frac{\Delta\theta}{\Delta t}$).

• Connection between linear and angular speed: $v=R \omega$.

Acceleration towards the Center (Centripetal Acceleration)

• Size (magnitude): $a_c=\frac{v^2}{R}=\omega^2 R=4\pi^2\nu^2 R$.

• Direction: Always points towards the center of the circle (inward). Even though the speed is constant, the direction of velocity is always changing, so there's always an acceleration.

Period & Frequency

• Period (T): The time it takes for one full circle: $T=\frac{2\pi R}{v}$.

• Frequency ($\nu$): How many circles it completes per second: $\nu=\frac{1}{T}$.

Worked Examples (Key Parts)

• Mistake with umbrella in rain and wind (Example 3.1): Helps figure out how to hold an umbrella. The actual velocity makes you hold it at an angle of $19^{\circ}$ east from straight up.

• How the law of cosines/sines for vectors is found (Example 3.2).

• Boat crossing a river with current (Example 3.3): The boat's actual speed is about $22\,\text{km/h}$, and it drifts about $23^{\circ}$ west of north.

• Motion changing over time (Example 3.4): If position is given by $\vec{r}(t)=3t\hat{i}+2t\hat{j}+5t^2\hat{k}$, then velocity is $\vec{v}(t)=3\hat{i}+2\hat{j}+10t\hat{k}$ and acceleration is $\vec{a}=10\hat{k}$. At $t=1\,\text{s}$, the speed is $5\,\text{m/s}$ at an angle of $53^{\circ}$ to the x-axis in the x-y plane.

• Simple motion in a flat space with steady acceleration (Example 3.5): Calculates position, finds y-coordinate is 36 m when x is 84 m at 6 s, and speed is 26 m/s.

• Projectile numbers (Examples 3.6–3.8): Shows real calculations for object launches and how complementary angles work.

• Bug on a spinning record (Example 3.9): Finds angular speed ($\omega=0.44\,\text{rad/s}$), linear speed ($v=5.3\,\text{cm/s}$), and acceleration ($a=2.3\,\text{cm/s}^2$).

Quick Look at Main Formulas

• Multiplying by numbers: $\lambda\vec{A}$.

• Adding vectors using their parts: $\vec{R}=(A<em>x+B</em>x)\hat{i}+(A<em>y+B</em>y)\hat{j}$.

• Motion with steady acceleration: $\vec{v}=\vec{v}<em>0+\vec{a}t$, and $\vec{r}=\vec{r}</em>0+\vec{v}_0t+\tfrac12\vec{a}t^2$.

• Projectile motion: Max height $h<em>m=\frac{v</em>0^{2}\sin^{2}\theta<em>0}{2g}$, Range $R=\frac{v</em>0^{2}\sin2\theta<em>0}{g}$, Total flight time $T</em>f=\frac{2v<em>0\sin\theta</em>0}{g}$.

• Circular motion (UCM): Acceleration towards center $a_c=\frac{v^2}{R}=\omega^2 R$, Speed related to angular speed $v=R\omega$, Angular speed related to frequency $\omega=2\pi \nu$.

Things to Think About (Important Ideas)

• Path length vs. displacement: They are only the same if you move in one straight line without changing direction.

• Average speed: It's always greater than or equal to the size of the average velocity.

• Vector equations: The equations for motion with vectors don't care how you set up your x and y axes; you can break them into parts later.

• UCM and constant acceleration: The equations for constant acceleration do not work for uniform circular motion because the direction of acceleration is always changing in UCM.

• Resultant vs. relative velocity: Don't confuse adding velocities ($\vec{v}<em>1+\vec{v}</em>2$) with finding how fast one object moves compared to another ($\vec{v}<em>{12}=\vec{v}</em>1-\vec{v}_2$).

• Centripetal direction: The acceleration points to the center only if the speed is constant.

• Same acceleration, different paths: Gravity provides the same acceleration for dropping an object or throwing it like a projectile, but their paths look very different because of their initial positions and velocities.

Exercises to Practice (Snapshot for Review)

• Telling apart scalars and vectors; understanding vector inequalities like ||\vec{A}+\vec{B}||<||\vec{A}|+||\vec{B}|| (the sum of two sides of a triangle is longer than the third side).

• Finding displacement for skaters on a round rink, a driver taking a winding path, or a taxi driver cheating on distance vs. actual displacement.

• Calculating limits for how far objects can be thrown into a ceiling or off a cliff; finding the maximum height of a cricket ball; speed when a stone hits the ground after being thrown off a cliff.

• Circular motion: comparing centripetal acceleration to gravity's pull; problems about circle size and how fast it spins.

• Breaking down vectors and finding their parts when the coordinate system