Change in position from initial to final.

$\Delta \vec{r} = \vec{r}_f - \vec{r}_i$

In one dimension: $\Delta x = x_f - x_i$

Units: meters (m)

Key points:

• Vector quantity (magnitude and direction)
• Dependent on coordinate system choice
• Can be positive or negative

Instantaneous velocity (at a specific instant): $\vec{v}(t) = \lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt}$

Average velocity (over time interval): $\bar{v} = \frac{\Delta \vec{r}}{\Delta t} = \frac{\vec{r}_f - \vec{r}_i}{\Delta t}$

Instantaneous acceleration (at a specific instant): $\vec{a}(t) = \lim_{\Delta t \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d\vec{v}}{dt}$

Average acceleration (over time interval): $\bar{a} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_f - \vec{v}_i}{\Delta t}$

Units: m/s and m/s2

Position vs. time graph.

• Slope = instantaneous velocity at that time
• Area = no direct physical meaning (not directly useful)
• Steepness indicates speed (steeper = faster)
• Positive slope: moving in +x direction
• Negative slope: moving in -x direction
• Horizontal line: object at rest

Key features:

• Curve represents changing velocity
• Straight line represents constant velocity
• Inflection point represents maximum/minimum acceleration

v-t Graphs:

Velocity vs. time graph.

• Slope = instantaneous acceleration at that time
• Area under curve = displacement (change in position)
• Height = velocity magnitude
• Above x-axis: positive displacement
• Below x-axis: negative displacement

Key features:

• Zero crossing: object changes direction (turnaround)
• Horizontal line: constant acceleration = 0
• Straight line: constant acceleration

Velocity vs. time graph.

• Slope = instantaneous acceleration at that time
• Area under curve = displacement (change in position)
• Height = velocity magnitude
• Above x-axis: positive displacement
• Below x-axis: negative displacement

Key features:

• Zero crossing: object changes direction (turnaround)
• Horizontal line: constant acceleration = 0
• Straight line: constant acceleration

Acceleration vs. time graph.

• Slope = rate of change of acceleration (jerk)
• Area under curve = change in velocity
• Height = acceleration magnitude

Key features:

• Often constant acceleration problems (horizontal line)
• Variable acceleration: integrate to find v and x

Slope interpretation:

• x-t graph slope -> velocity
• v-t graph slope -> acceleration
• a-t graph slope -> rate of change of acceleration (jerk)

Area under curve interpretation:

• v-t graph area -> displacement
• a-t graph area -> change in velocity
• Integration of slope -> corresponding quantity

Calculus connections:

• Slope = derivative
• Area = integral
• Fundamental theorem of calculus connects them

x-t graph:

• Position vs. time
• Steeper slope = faster motion
• Turning points = velocity zero (direction change)
• Curvature = acceleration

v-t graph:

• Velocity vs. time
• Above axis = positive velocity (moving forward)
• Below axis = negative velocity (moving backward)
• Zero crossings = direction changes
• Area = displacement

a-t graph:

• Acceleration vs. time
• Positive = speeding up in + direction
• Negative = slowing down or moving backward faster
• Area = velocity change

Instantaneous velocity equals derivative of position with respect to time.

$\vec{v}(t) = \frac{d\vec{r}}{dt}$

In one dimension: $v(t) = \frac{dx}{dt}$

Physical meaning:

• Rate of change of position
• Slope of position-time graph at each point

Example: If $x(t) = 3t^2 + 2t$, then $v(t) = 6t + 2$

Instantaneous acceleration equals derivative of velocity with respect to time.

$\vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}$

In one dimension: $a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2}$

Physical meaning:

• Rate of change of velocity
• Slope of velocity-time graph
• Second derivative of position

Example: If $v(t) = 6t + 2$, then $a(t) = 6$ (constant)

Deltax = ∫v dt:

Displacement equals integral of velocity over time.

$\Delta x = \int_{t_1}^{t_2} v(t) \, dt$

From calculus definition: $x(t) = x_0 + \int_0^t v(t') \, dt'$

Physical meaning:

• Area under velocity-time graph
• Accumulation of velocity over time

Example: If $v(t) = v_0 - gt$ (upward motion), then $\Delta x = v_0t - \frac{1}{2}gt^2$

Deltav = ∫a dt:

Change in velocity equals integral of acceleration over time.

$\Delta v = \int_{t_1}^{t_2} a(t) \, dt$

From calculus definition: $v(t) = v_0 + \int_0^t a(t') \, dt'$

Physical meaning:

• Area under acceleration-time graph
• Accumulation of acceleration over time

Example: If $a(t) = -g$ (free fall), then $\Delta v = -gt$ and $v(t) = v_0 - gt$

Kinematics quantities are vectors.

Vector addition for displacements: $\Delta \vec{r} = \Delta \vec{r}_1 + \Delta \vec{r}_2 + ...$

Independence of perpendicular components:

• x and y motions are independent
• Motion in each dimension described separately
• Combine using vector addition

2D kinematics: $\vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2}\vec{a} t^2$

Velocity of object A relative to object B:

$\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$

Chain rule (velocity of A relative to C): $\vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC}$

Applications:

• Ships/planes moving on moving Earth's surface
• Collision problems in different reference frames
• River boat problems

Galilean transformation (for relative velocities much less than c): $\vec{v}' = \vec{v} - \vec{v}_{frame}$

Motion of object launched at angle, under gravity only.

Assumptions:

• Neglect air resistance
• Constant acceleration: $\vec{a} = -g\hat{j}$ (downward)
• Horizontal acceleration: $a_x = 0$
• Vertical acceleration: $a_y = -g$

Equations:

• x-direction (constant velocity): $x = v_0\cos\theta \cdot t$ $v_x = v_0\cos\theta$

• y-direction (constant acceleration): $y = v_0\sin\theta \cdot t - \frac{1}{2}gt^2$ $v_y = v_0\sin\theta - gt$

Time of flight (return to launch height): $T = \frac{2v_0\sin\theta}{g}$

Maximum range (level ground): $R = \frac{v_0^2\sin(2\theta)}{g}$

Maximum at $\theta = 45 degrees$: $R_{max} = \frac{v_0^2}{g}$

Maximum height: $H = \frac{(v_0\sin\theta)^2}{2g}$

Trajectory equation (parabolic path): $y = x\tan\theta - \frac{gx^2}{2v_0^2\cos^2\theta}$