chapter 3 two-dimensional kinematics

### key concepts:

- motion concepts:

- all two-dimensional motion can be broken into horizontal & vertical components

- these components are independent of each other

#### practice problem!

walking in a city:

when walking nine blocks east then five blocks north

- total distance walked: 14 blocks

- straight-line distance: 10.3 blocks $(\sqrt{9^2+5^2})$

- the straight-line path forms the hypotenuse of a right triangle

below is a demonstration how vector addition works in two-dimensional motion:

![[Pasted image 20260206110630.png]]

## vector addition & subtraction: graphs

### how are vectors represented in graphs & problems??

- bold symbols represent vectors (e.g., *D**)*

- italic symbols represent magnitude (e.g., D)

- arrows show DIRECTION, length or vectors show MAGNITUDE

### what the heck is the head-to-tail method??

the graphical method for adding vectors:

1. draw the first vector with a proper magnitude & direction

2. place the tail of the second vector at the head of the first vector

3. do the same for however many vectors you have

4. draw the resultant from the tail of the first vector to the head of the last vector

5. measure the magnitude with a ruler & direction with a protractor

![[Pasted image 20260206111319.png]]

### how am i supposed to subtract vectors???

- $A\space - \space B = A \space + \space (-B)$

- negative vector: same magnitude, opposite direction

- subtraction is commutative: order doesn't affect the result

## vector addition and subtraction: analytical

### mathematical approach

instead of using the methods we used on graphs, we will be using:

- trigonometric relationships for direction

- pythagorean theorem for magnitude

for perpendicular vectors:

- resultant magnitude: $R=\sqrt{A^2+B^2}$

- direction: $\theta=\tan^{-1} (\frac{B}{A})$

## projectile motion

### properties of projectiles

- acceleration: constant downward due to gravity $(g = 9.8\space m/s^2)$

- trajectory: curved path (usually a parabola)

- range: horizontal distance the object traveled

- maximum height: the highest point in the trajectory

### what's the key principle???

- the motion of a projectile can be analyzed by separating it into horizontal & vertical components that are independent of each other

- horizontal motion: has a constant velocity (no acceleration)

- vertical motion: has a constant acceleration due to gravity

![[Pasted image 20260206120421.png]]

## addition of velocities

### relative velocity

velocity depends on the observer's frame of reference. when adding velocities:

- use the vector addition principles

- consider the relative motion between objects

#### okay but what the heck is the formula???

$\vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC}$

where:

$\vec{v}_{AC}$ = velocity of A relative to C

$\vec{v}_{AB}$ = velocity of A relative to B

$\vec{v}_{BC}$ = velocity of B relative to C

## summary table:

| concept | key formula | key principle |

| ---------------------------------- | -------------------------------------------- | ---------------------------------------------------------------- |

| resultant of perpendicular vectors | $R=\sqrt{A^2\space + \space B^2}$ | pythagorean theorem |

| vector direction | $\theta = \tan^{-1}(\frac{B}{A})$ | trigonometry |

| projectile motion | independent components | horizontal: constant velocity<br>vertical: constant acceleration |

| relative velocity | $\vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC}$ | vector addition |

## vector multiplication & scalar operations

### vector-scalar multiplication

when multiplying a vector by a scalar:

- magnitude changes: magnitude becomes the absolute value of the scalar $|scalar|$

- direction **remains the same** IF scalar is positive

- direction reverses if scalar is negative

for example, multiplying a vector by three would triple its magnitude but keeps the direction. multiplying by negative two would double the magnitude & reverses direction

### division by scalars

division is equivalent to multiplication by the reciprocal:

- divide by $2$ = multiplying by $\frac{1}{2}$

- same rules apply as scalar multiplication

## resolving vectors into components

### perpendicular components

any vector can be broken into perpendicular components (typically x and y axes). this is the inverse of vector addition--find what vectors add together to produce the original

![[Pasted image 20260206123719.png]]

### component equations

for a vector $A$ at angle $\theta$:

- x-component: $A_x=A\cos \theta$

- y-component: $A_y=A\sin\theta$

- magnitude: $A=\sqrt{A^2_x+A^2_y}$

- direction: $\theta=\tan^{-1}(\frac{A_y}{A_x})$

key principle: components along the same axis can be added like ordinary numbers because they lie along the same line

![[Pasted image 20260206132544.png]]

## analytical methods for vector operations

### advantages over graphical methods

- more precise: not limited by drawing accuracy

- more concise: uses geometry & trigonometry

- more accurate: limited only by measurement precision

#### step-by-step analytical addition

step 1: find components of each vector along chosen axes

- use $A_x=|\vec A|\cos\theta$ and $A_y=|\vec A|\sin\theta$

step 2: add components along each axis separately

- $R_x=A_x+B_x$

- $R_y=A_y+B_y$

step 3: find the magnitude of resultant using pythagorean theorem

- $R=\sqrt{R^2_x\space + \space R^2_y}$

step 4: find the direction of the resultant

- $\theta=\tan^{-1}(\frac{R_y}{R_x})$

![[Pasted image 20260206133332.png]]

## projectile motion analysis

### defining characteristics

- acceleration: only due to gravity ($a_y=-g=-9.8{\space}m/s^2)$

- horizontal acceleration: $a_x=0$ (neglecting air resistance)

- trajectory: parabolic path

### four-step analysis method

- step 1: resolve motion into horizontal (x) and vertical (y) components

- initial velocity components: $v_{0x}=v_0\cos\theta\space , \space v_{0y}=\sin\theta$

- step 2: treat as two independent 1D motions

- horizontal: constant velocity motion

- vertical: constant acceleration motion

- step 3: solve for unknowns in each motion separately

- time is the common variable between both motions

- step 4: recombine motions to find total displacement & velocity

- use vector addition techniques

### key equations

- horizontal motion: $x=x_0+v_{0x}t$

- vertical motion: $y=y_0+v_{oy}t-\frac{1}{2}gt^2$

- velocity components: $v_x=v_{0x}\space , \space v_y=v_{0y}-gt$

### practice problem!

given:

vector A: 53.0 m at 20.0$\degree$

vector B: 34.0 m at 63.0$\degree$

solution process:

1. add x-components:

- $R_x=(A\times\cos(20\degree)+(B\times\cos(20\degree))=(49.8)+(15.4)=65.2$

2. add y-components:

- $R_y=(A\times \sin(20))+(B\times sin(63))=(18.1)+(30.3)=48.4$

3. calculate resultant magnitude:

- $R=\sqrt{R^2_x+R^2_y}=\sqrt{(65.2)^2+(48.4)^2}=\sqrt{(4251.04)+(2342.56)}=\sqrt{6593.6}=81.2$

4. calculate direction:

- $\theta=\tan^{-1}(\frac{R_y}{R_x})=\tan^{-1}(\frac{48.4}{65.2})=36.6\degree$

5. result

- $\vec{R}=81.2\space m \space at \space 36.6\degree north \space of \space east$

![[Pasted image 20260206214821.png]]

## key takeaways

| concept | formula | when to use |

| ------------------- | ---------------------------------------------- | ----------------------------------------- |

| vectors components | $A_x=A\cos\theta,\space A_y=A\sin\theta$ | breaking vectors into perpendicular parts |

| resultant magnitude | $R\sqrt{R^2_x+R^2_y}$ | finding total magnitude from components |

| resultant direction | $\theta=\tan^{-1}(\frac{R_y}{R_x})$ | finding direction from components |

| projectile motion | $x=v_0t\space , \space y=v_0t-\frac{1}{2}gt^2$ | analyzing 2D projectile motion |

## projectile range & orbital motion

### factors affecting projectile range

building on our understanding of projectile motion, the range of a projectile depends on several key factors:

- initial speed effects:

- greater initial speed produces greater range for given angle

- range increases proportionally w/ launch velocity

- launch angle effects:

- for a fixed initial speed, the maximum range occurs at $\theta = 45\degree$ (neglecting air resistance)

- with air resistance, the optimal angle is approximately $38\degree -40\degree$

- for every angle except $45\degree$, there are two angles that give the same range

- these angle pairs satisfy: $\theta_1 +\theta_2 = 90\degree$

### range equation

the range of a projectile on level ground with negligible air resistance is:

$R=\frac{v^2_0\sin (2\times\theta)}{g}$

where:

- $R$ = horizontal range

- $v_0$ = initial speed

- $\theta$ = launch angle

- $g$ = acceleration due to gravity

### gravity's effect on range this explains why:

- golf balls travel farther on the moon (weaker gravity)

- athletes can jump higher on the moon with the same effort

### orbital motion connection

when projectile range becomes very large:

- the earth curves away below the projectile

- gravity changes direction along the path

- the projectile has farther to fall than on level ground

- with sufficient initial speed, the projectile achieves orbit

- orbital principle: the object falls continuously but never hits the surface because the earth curves away at the same rate as it falls

## relative velocity applications

### understanding relative motion

definition: relative velocity is the velocity of an object as measured from a particular reference frame

key principle: velocities must be specified relative to reference frame, and different observers may measure dramatically different velocities

### vector addition of velocities

the general equation for combining velocities is:

$\vec{v}_{AC} = \vec{v}_{AB}+\vec{v}_{BC}$

where:

- $\vec{v}_{AC}$ = velocity of A relative to C

- $\vec{v}_{AB}$ = velocity of A relative to B

- $\vec{v}_{BC}$ = velocity of B relative to C

### practical examples

boat crossing a river:

- boat moves perpendicular to shore at $0.75\space\frac{m}{s}$

- river flows parallel to shore at $1.20\space\frac{m}{s}$

- resultant velocity is relative to shore:

- $v=\sqrt{(0.75)^2+(1.20)^2}=1.42\space\frac{m}{s}$

- $\theta=\tan^{-1}(\frac{1.20}{0.75})=58.0\degree$

airplane in crosswind:

- plane flies north at $45.0\space\frac{m}{s}$ relative to air

- ground track shows $38.0\space\frac{m}{s}$ at angle west of north

- wind velocity can be calculated using vector components

## classical relativity

### definition & scope

classical relativity studies relative velocities when speeds are less than 1% of light speed *(about $3\space 000\frac{m}{s}$)

### galileo's ship thought experiment

when binoculars are dropped from a ship's mast:

- observer on ship: sees vertical fall straight down

- observer on shore: sees parabolic trajectory

- both observers: see the binoculars hit the deck at the same location

classical relativity principle: all observers agree on the final outcome, but describe the motion differently based on their reference frame

### modern extensions

this foundation led to einstein's theory of relativity, which applies when velocities approach the speed of light

## summary of key principles

| concept | formula | application |

| ------------------------ | ---------------------------------------------- | ------------------------ |

| maximum projectile range | best at $\theta = 45\degree$ | sports, ballistics |

| range equation | $R=\frac{v^2_0\sin(2\times\theta)}{g}$ | calculate distance |

| velocity addition | $\vec{v}_{AC}=\vec{v}_{AB}+\vec{v}_{BC}$ | relative motion problems |

| velocity components | $v_x=v\cos\theta\space ,$<br>$v_y=v\sin\theta$ | analyze 2D motion |

| orbital condition | projectile speed = orbital speed | satellite motion |

### problem-solving strategy

1. identify reference frames for each object

2. choose coordinate system aligned with motion components

3. apply vector addition using components

4. calculate resultant magnitude and direction