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Physics · Kinematics

Motion &
Acceleration

Uniform & Non-Uniform — equations, simulations, graphs, data

Variables
uInitial velocity (m/s)
vFinal velocity (m/s)
aAcceleration (m/s²)
tTime (s)
sDisplacement (m)
The Four Equations
EQ. 01
v = u + at
Velocity–Time Relation
Final velocity equals initial velocity plus the product of acceleration and time. Direct consequence of the definition of acceleration.
v = final vel. u = initial vel. a = accel. t = time
EQ. 02
s = ut + ½at²
Displacement–Time Relation
Displacement is the area under the v–t graph: a rectangle (ut) plus a triangle (½at²). Integrating velocity over time.
s = displacement ½at² = accel. component
EQ. 03
v² = u² + 2as
Velocity–Displacement Relation
Eliminates time. Derived by combining Eq. 1 and Eq. 2. Extremely useful when time is unknown.
no time needed energy analogy
EQ. 04
s = (u+v)/2 · t
Average Velocity Relation
Displacement equals average velocity multiplied by time. Works because velocity changes linearly under uniform acceleration.
avg. vel. = (u+v)/2 linear change
Live Car Simulation
5 m/s
3 m/s²
4 s
▶ ROAD VIEW t = 0.0 s
LIVE GRAPHS
VELOCITY (m/s)
DISTANCE (m)
ACCEL (m/s²)
DIST
0.0
VEL
0.0
VELOCITY (v)
—
m/s
DISPLACEMENT (s)
—
m
TIME ELAPSED
—
s
AVG VELOCITY
—
m/s
Motion Graphs
VELOCITY vs TIME  v = u + at
DISPLACEMENT vs TIME  s = ut + ½at²
ACCELERATION vs TIME  a = const.
Live Data Table
Time (s) Velocity v (m/s) Displacement s (m) Acceleration a (m/s²)
Run the simulation to populate data
Derivation with Car Proof
01
Definition: a = (v − u) / t
Acceleration is the rate of change of velocity. If a car changes from u to v in time t, then a = (v − u) / t.
02
Rearrange → v = u + at [Eq. 1]
Multiply both sides by t: at = v − u, then add u: v = u + at. 🚗 A car starting at 5 m/s with a = 3 m/s² reaches v = 5 + 3×4 = 17 m/s after 4 s.
03
Average velocity = (u + v) / 2
Under uniform acceleration, velocity increases linearly, so the average is exactly the midpoint. 🚗 Same car: avg = (5 + 17) / 2 = 11 m/s.
04
s = avg vel × t = (u+v)/2 · t [Eq. 4]
Distance = average velocity × time. 🚗 s = 11 × 4 = 44 m.
05
Substitute v = u + at into Eq. 4:
s = (u + u + at)/2 · t = (2u + at)/2 · t = ut + ½at² → Eq. 2 ✓
06
From Eq.1: t = (v−u)/a. Sub into Eq.4:
s = (u+v)/2 · (v−u)/a = (v²−u²)/(2a). Rearranging: v² = u² + 2as [Eq. 3] ✓
🚗 Car proof: v² = 5² + 2×3×44 = 25 + 264 = 289, so v = √289 = 17 m/s ✓
v² = u² + 2as ✓

All three derived values agree: the car reaches 17 m/s covering 44 m in 4 s.
Every equation is consistent — this is the beauty of uniform acceleration.

Non-Uniform Acceleration

In non-uniform acceleration, the rate of change of velocity is not constant. The car speeds up and slows down irregularly. Acceleration changes with time: a(t) = a₀·sin(ωt) + k·t — a blend of sinusoidal and linear variation. Velocity and displacement are computed by numerical integration (Euler method).

uInitial velocity (m/s)
a₀Accel. amplitude (m/s²)
ωAngular freq (rad/s)
kLinear accel. growth
TDuration (s)
Live Car Simulation
5 m/s
4 m/s²
0.5 r/s
0.20
8 s
▶ ROAD VIEW t = 0.0 s
LIVE GRAPHS
VELOCITY (m/s)
DISTANCE (m)
ACCEL (m/s²)
DIST
0.0
VEL
0.0
VELOCITY (v)
—
m/s
DISPLACEMENT (s)
—
m
TIME ELAPSED
—
s
ACCELERATION (a)
—
m/s²
Top-View 2D Spiral Track LAP 0

Watch the car spiral outward on a 2D Archimedean path. Trail rays mark every metre travelled. A distance bar floats above the car head, and milestone markers are stamped on the path.

▶ TOP VIEW — SPIRAL PATH t = 0.0 s
0.25× SLOW-MO
LIVE GRAPHS
VELOCITY (m/s)
DISTANCE (m)
ACCEL (m/s²)
DIST
0.0
VEL
0.0
Motion Graphs
VELOCITY vs TIME  v = ∫a(t)dt + u
DISPLACEMENT vs TIME  s = ∫v(t)dt
ACCELERATION vs TIME  a(t) = a₀·sin(ωt)+kt
Live Data Table
Time (s) Velocity v (m/s) Displacement s (m) Acceleration a (m/s²)
Run the simulation to populate data
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