• Scalar: Quantity with magnitude only (no direction)
• Examples: distance, speed, mass, time, energy
• Added/subtracted using ordinary arithmetic
• Vector: Quantity with both magnitude and direction
• Examples: displacement, velocity, acceleration, force, momentum
• Must consider direction in addition/subtraction
• Vectors are denoted with arrow above symbol ($\vec{v}$) or boldface ($\textbf{v}$)

• Graphical method: Tip-to-tail method
• Addition: Place tail of second vector at tip of first
• Resultant: From tail of first to tip of last
• Subtraction: Add the negative of the second vector
• Component method: Break vectors into x and y components
• Add/subtract corresponding components
• Reconstruct resultant from components

• $v_x = v\cos\theta$ (horizontal component)
• $v_y = v\sin\theta$ (vertical component)
• $v = \sqrt{v_x^2 + v_y^2}$
• $\theta = \tan^{-1}(v_y/v_x)$
• Essential for 2D motion problems

• Vectors with magnitude of 1
• $\hat{i}$: unit vector in x-direction
• $\hat{j}$: unit vector in y-direction
• $\vec{v} = v_x\hat{i} + v_y\hat{j}$
• Used to specify direction of vectors

• Distance (scalar): Total path length traveled
• Always positive
• Path-dependent
• Displacement (vector): Change in position
• Can be positive, negative, or zero
• Straight line from initial to final position
• $\Delta x = x_f - x_i$

• $\bar{v} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$
• Vector quantity (direction matters)
• Can be zero even if average speed is not
• Different from instantaneous velocity

• $v = \lim_{\Delta t \to 0}\frac{\Delta x}{\Delta t} = \frac{dx}{dt}$
• Velocity at a specific instant
• Slope of position-time graph at that point
• Tangent to the curve at that point

• $a = \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$
• Rate of change of velocity
• Vector quantity (can have direction different from velocity)
• Units: m/s2
• Can occur while speeding up, slowing down, or turning

• Slope = instantaneous velocity
• Steeper slope = higher velocity
• Horizontal line = zero velocity (at rest)
• Curved line = changing velocity (acceleration present)
• Positive slope = positive velocity (moving forward)

#VALUE!

• Area under v-t graph = displacement
• Area above time axis = positive displacement
• Area below time axis = negative displacement
• Useful for calculating displacement with changing velocity

• Area under a-t graph = change in velocity
• $\Delta v = \int a \, dt$
• Can determine final velocity from acceleration-time graph

• Series of dots showing object's position at equal time intervals
• Closer dots = slower speed
• Farther apart dots = faster speed
• Equal spacing = constant velocity
• Direction indicated by arrows

• $\vec{v}_{A/C} = \vec{v}_{A/B} + \vec{v}_{B/C}$
• Example: Boat crossing river with current
• Essential for relative motion problems
• Vector addition required

• Reference frame moving at constant velocity (or at rest)
• Newton's laws hold in inertial frames
• No fictitious forces needed
• Non-inertial frames: accelerating frames require fictitious forces

Constant Acceleration Equations / UAM:

• $v = v_0 + at$
• $x = x_0 + v_0t + \frac{1}{2}at^2$
• $v^2 = v_0^2 + 2a\Delta x$
• $\bar{v} = \frac{v + v_0}{2}$ (valid only for constant acceleration)
• Apply separately to x and y components for 2D motion

• Special case of constant acceleration with $a = g = 9.8$ m/s2 downward
• Objects dropped: $v_0 = 0$
• Objects thrown upward: $a = -g$ (deceleration until peak)
• Time to reach peak: $t_{up} = v_0/g$
• Maximum height: $h_{max} = v_0^2/(2g)$
• Round trip time: $t_{total} = 2v_0/g$

• $g = 9.8$ m/s2 near Earth's surface
• Varies with altitude and planet
• Independent of mass (Galileo's principle)
• All objects fall with same acceleration in vacuum

• Horizontal and vertical motions are independent
• Horizontal: constant velocity (assuming no air resistance)
• Vertical: constant acceleration ($a_y = -g$)
• Same time applies to both motions
• Problems solved by treating components separately

• Horizontal: $x = v_0 t$ (constant velocity)
• Vertical: $y = h - \frac{1}{2}gt^2$ (free fall from height h)
• Path: parabolic
• Time to hit ground: $t = \sqrt{2h/g}$
• Range: $R = v_0\sqrt{2h/g}$

• Initial velocity components:
• $v_{0x} = v_0\cos\theta$
• $v_{0y} = v_0\sin\theta$
• Position:
• $x = v_0\cos\theta \cdot t$
• $y = v_0\sin\theta \cdot t - \frac{1}{2}gt^2$
• Velocity:
• $v_x = v_0\cos\theta$ (constant)
• $v_y = v_0\sin\theta - gt$
• Maximum Height: $h_{max} = \frac{(v_0\sin\theta)^2}{2g}$
• Time to max height: $t_{up} = \frac{v_0\sin\theta}{g}$
• Total time: $t_{total} = \frac{2v_0\sin\theta}{g}$
• Range: $R = \frac{v_0^2\sin(2\theta)}{g}$
• Maximum range at $\theta = 45^\circ$ (for symmetric launch and landing)

Key projectile formulas for angled launch.

• Maximum Height: $h_{max} = \frac{(v_0\sin\theta)^2}{2g}$
• Time to max height: $t_{up} = \frac{v_0\sin\theta}{g}$
• Total time: $t_{total} = \frac{2v_0\sin\theta}{g}$
• Range: $R = \frac{v_0^2\sin(2\theta)}{g}$
• Maximum range at $\theta = 45^\circ$ (for symmetric launch and landing)

Independence of Perpendicular Motions:

• Horizontal and vertical motions are independent
• Horizontal: constant velocity (assuming no air resistance)
• Vertical: constant acceleration ($a_y = -g$)
• Same time applies to both motions
• Problems solved by treating components separately