Uniformly Accelerated Motion

Uniformly accelerated motion is when velocity changes by the same amount each second, like a car pulling away after a green light. With constant acceleration the v-t graph is a straight line: slope is acceleration and area under it is displacement. Three linked formulas for velocity, displacement, and time cover free fall and many textbook cases. Slide initial speed and acceleration to watch the v-t graph and freefall spacing change.

Motion with Steadily Changing Velocity

🏎️ Intuition for Uniform Acceleration

①A car starting after a green light — its speed keeps increasing

②With constant acceleration, velocity changes by the same amount each second

③a = 3 m/s² means velocity increases 3 m/s per second!

Reading Uniform Acceleration on v-t Graph

Initial Velocity v₀2 m/s

Acceleration a3 m/s²

📊 Graph Interpretation

①Straight v-t line = uniform acceleration

②Slope = acceleration a (positive: speeding up, negative: slowing down)

③Area under graph = displacement s (trapezoidal area)

The Three Equations of Uniform Acceleration

Velocity Equation

v = v₀ + at

final velocity = initial velocity + acceleration × time

Displacement Equation

s = v₀t + 12at²

displacement = v₀ × t + ½ × a × t²

Velocity–Displacement Equation

v² = v₀² + 2as

Relation between velocity and displacement without t

💡 Choosing the Right Equation

①If s is unknown → use v = v₀ + at

②If v is unknown → use s = v₀t + ½at²

③If t is unknown → use v² = v₀² + 2as

Free Fall: The Quintessential Uniform Acceleration

Free-Fall Equation

v = gt, s = 12gt² (g ≈ 9.8 m/s²)

Initial velocity v₀ = 0, acceleration a = g (gravitational acceleration)

Worked Examples

Example 1

An object with initial speed 5 m/s accelerates uniformly at 2 m/s². What is its speed after 4 s?

Use the velocity equation v = v₀ + at.

v = v₀ + at

Substitute v₀ = 5, a = 2, t = 4.

v = 5 + 2 × 4 = 13 m/s

▸ 13 m/s

To find final speed from time and acceleration, use v = v₀ + at — the simplest equation, no displacement needed.

Example 2

An object is released from rest and falls freely for 3 s. How far does it fall? (g = 10 m/s²)

Free fall is uniformly accelerated motion with v₀ = 0, so use s = ½gt².

s = 12gt²

Substitute g = 10, t = 3.

s = 12 × 10 × 3² = 45 m

▸ 45 m

Free fall is uniform acceleration with v₀ = 0, a = g. The distance fallen is proportional to time squared.

Summary

Velocity

v = v₀ + at

Displacement

s = v₀t + 12at²

Velocity–Displacement

v² = v₀² + 2as

Equation eliminating time

CSAT-style

A car moving at 20 m/s on a straight road brakes and stops after travelling 40 m more. What is its acceleration?

①−2 m/s²

②−4 m/s²

③−5 m/s²

④−8 m/s²

⑤−10 m/s²

▸ ③ −5 m/s²

Time t is unknown, so use v² = v₀² + 2as (stops ⇒ v = 0).

0 = v₀² + 2as

Substitute v₀ = 20, s = 40 to solve for a.

400 + 80a = 0 ⇒ a = −5 m/s²

🎯 Exam Points

①Memorize the three equations: v=v₀+at, s=v₀t+½at², v²=v₀²+2as

②v-t graph: slope = acceleration, area = displacement

③Free fall: substitute v₀=0, a=g=9.8 m/s²

④Vertical throw upward: v₀>0, a=-g (gravity decelerates)

⑤Average velocity = (v₀+v)/2 (only for uniform acceleration)

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Uniform Linear Motion

Newton's Laws

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