Kinematics in One Dimension – Quick‐Reference Notes

Motion & Reference Frames

• Motion = change in position as a function of time; always stated relative to a chosen frame of reference.

• Same event can appear as “moving” or “at rest” to different observers → always specify reference point.

Scalars vs. Vectors

• Scalars: magnitude only (e.g., mass, temperature, distance, speed).

• Vectors: magnitude + direction (e.g., displacement, velocity, acceleration, force).

Kinematic Variables

• Distance: actual path length (scalar, m).

• Displacement $\Delta x$: straight-line change in position (vector, m; can be ±).

• Time $t$: interval between events (s).

• Speed $s = \frac{\text{distance}}{t}$ (scalar, m/s).

• Velocity $v = \frac{\Delta x}{\Delta t}$ (vector, m/s; sign gives direction).

• Acceleration $a = \frac{\Delta v}{\Delta t}$ (vector, m/s$^2$; + increasing speed in +x, − decreasing, etc.).

Graphical Interpretation

• Position–Time graph:
– Slope $m = \frac{\Delta y}{\Delta x} = \frac{\Delta x}{\Delta t} = v$.
• Velocity–Time graph:
– Slope $m = \frac{\Delta v}{\Delta t} = a$.
– Area under curve $= v \times t = \Delta x$ (displacement).
• Acceleration–Time graph:
– Area $= a \times t = \Delta v$ (change in velocity).

Average vs. Instantaneous

• Average velocity $\bar v = \frac{\Delta x}{\Delta t}$ over interval.

• Instantaneous velocity $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$ (slope at a point).

• Same idea for acceleration.

Constant Acceleration (Uniformly Accelerated Motion)

If $a = \text{constant}$, the five variables ${\Delta x,\ t,\ v_0,\ v,\ a}$ are related by four kinematic formulas:

1. $v = v_0 + a t$

2. $x = x<em>0 + v</em>0 t + \tfrac12 a t^2$

3. $v^2 = v<em>0^2 + 2 a (x - x</em>0)$

4. $x = x<em>0 + \tfrac12 (v + v</em>0) t$
   Choose the equation containing the unknown plus three known quantities; valid only if $a$ is constant and all variables refer to same axis.

Free-Fall Motion

• Free fall: motion under gravity alone.

• Acceleration magnitude $g \approx 9.8\,\text{m/s}^2$ near Earth’s surface; always downward → use $a_y = -g$ if +y is upward.

• Same kinematic equations apply with $a = -g$.

• Object moving upward still experiences downward $g$; at top of flight $v=0$ but $a = -g$ (not zero).

Sign Conventions & Interpretation

• Positive/negative signs come from chosen axis; keep consistent.

• Zero acceleration ⇒ constant velocity (could be zero or non-zero).

• Non-zero acceleration can exist while object is momentarily at rest (e.g., turning point in projectile).

Quick Problem-Solving Strategy

1. Draw diagram; set coordinate axis.

2. List knowns/unknowns.

3. Decide if acceleration is constant; pick proper kinematic formula.

4. Apply sign convention carefully.

5. Check units and reasonableness of answer.

Graphing Tasks (Summary)

• Plot $x$ vs $t$; slope gives constant velocity.

• Compute $v$ for each interval and plot $v$ vs $t$; slope gives constant $a$.

• From $v$ values, compute $a$ and plot $a$ vs $t$ (horizontal line if $a$ constant).

Key Takeaways

• Motion description combines qualitative concepts and quantitative math.

• Graphs offer visual & numerical extraction of $v$, $a$, and $\Delta x$.

• Uniform acceleration links five variables via four equations; free-fall is special uniform-accel case with $a=-g$.