Kinematics (1D) – Formula Sheet
Key definitions, equations of motion, graphs, and quick exam notes (JEE Main).
1) Core quantities (1D)
- Position (x): location on a line (with sign).
- Displacement (Δx): change in position. vector in general
- Distance: total path length (always ≥ 0).
- Speed: rate of change of distance (scalar).
- Velocity (v): rate of change of displacement (can be +/−).
- Acceleration (a): rate of change of velocity (can be +/−).
Average velocity: v̄ = Δx / Δt
Average speed: speed̄ = (total distance) / Δt
Instantaneous velocity: v = dx/dt
Instantaneous acceleration: a = dv/dt = d²x/dt²
In 1D, signs matter. If v and a have opposite signs, speed decreases.
2) Uniform acceleration (most-used formulas)
v = u + at
s = ut + (1/2)at²
v² = u² + 2as
s = ((u + v)/2) t
- Relative separation (same line): If two particles A, B move on same line, Δx_AB = x_A − x_B, v_AB = v_A − v_B, a_AB = a_A − a_B.
- Equal acceleration case: If a_A = a_B, relative acceleration is 0 → relative velocity stays constant.
3) Graphs (JEE favourites)
| Graph | Key interpretation | Instant notes |
|---|---|---|
| x–t (position–time) |
Slope at a point = instantaneous velocity v Steeper slope ⇒ higher |v| |
Curving upward (slope increasing) ⇒ acceleration in same direction as velocity. Horizontal line ⇒ v = 0 (at rest). |
| v–t (velocity–time) |
Slope = acceleration a Area under curve = displacement s |
Line above time axis ⇒ positive v; below ⇒ negative v. Area gives signed displacement (not distance). |
| a–t (acceleration–time) | Area under curve = change in velocity Δv |
If a is constant ⇒ horizontal line. Use Δv to connect to v–t quickly. |
4) Free fall / vertical motion (take upward as +)
a = −g (g ≈ 9.8 m/s², often 10 m/s² in JEE)
v = u − gt
y = ut − (1/2)gt²
v² = u² − 2gy
- At highest point: v = 0, but a = −g (still).
- Time up = time down (if returns to same height, no air resistance).
5) Quick “avoid mistakes” checklist
- Use displacement (signed) for kinematics equations, not distance.
- Keep a consistent sign convention (right/up = + is common).
- For distance from v–t graph: add absolute areas (area magnitude), not signed area.
- If motion reverses direction, split the time interval into parts.
- Units: x in m, t in s, v in m/s, a in m/s².
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Last modified: December 14, 2025
