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Engine 3 of 7

UAM Master Engine

Uniformly Accelerated Motion • The Core of Kinematics

Purpose

Uniformly Accelerated Motion (UAM) is the most important model in Motion in a Straight Line.

Almost every NEET and JEE Main problem eventually reduces to UAM.

The four famous kinematic equations are not independent formulas.

One Assumption
↓
Constant Acceleration
↓
Four Equations

3.1 UAM Recognition Box

Before using any UAM formula, verify the following conditions.

✅ Acceleration is constant

✅ Motion is along a straight line

✅ Classical speeds

Trigger Words

a = 2 m/s²
a = -5 m/s²
g = 10 m/s²
Entire UAM Toolkit Available
Formula Misuse Alert

If acceleration is:

a = 2t
a = kx

UAM is invalid.

Use the Calculus Engine.

3.2 UAM Family Tree

a = constant
↓
Integration
↓
v = u + at

s = ut + ½at²

v² = u² + 2as

s = ½(u+v)t
If you forget a formula, derive it from the others.

Never memorize blindly.

3.3 Formula 1

Velocity–Time Equation

$$v=u+at$$

What It Does

Connects:

Velocity ↔ Time

Tells how velocity changes after time t.

Formula DNA

$$a=\frac{v-u}{t}$$

Rearranging gives:

$$v=u+at$$

Eliminates

Displacement (s)

Use When (Triggers)

Question Says Immediate Thought
Starts from rest u = 0
Comes to stop v = 0
Velocity after t sec Use v = u + at
Speed after time t Use v = u + at
Prohibited Situations

a = f(t)

a = f(x)

Circular Motion

UAM formula is not valid.
Lightning Example

u = 0
a = 3 m/s²
t = 5 s

$$v=u+at$$

$$v=15\;m/s$$

3.4 Formula 2

Displacement–Time Equation

$$s=ut+\frac12at^2$$

What It Does

Displacement ↔ Time

Formula DNA

Obtained by integrating velocity.

Eliminates

Final Velocity (v)

Use When

Question Says Think
Position after t sec Use this formula
Displacement after t sec Use this formula
How far after t sec Use this formula
Prohibited Situations

Need total distance

Direction reversal present

Variable acceleration
Killer Trap

Question asks:

Total distance travelled

Check first:

Did the particle reverse direction?

If YES,

Distance ≠ Displacement

3.5 Formula 3

Velocity–Displacement Equation

$$v^2=u^2+2as$$

What It Does

Velocity ↔ Displacement

Formula DNA

Obtained by eliminating time.

Eliminates

Time (t)

Master Trigger

Time Missing
↓
Think
↓
$$v^2=u^2+2as$$

Use When

Question Says Think
Braking distance Use this formula
After travelling s metres Use this formula
Maximum height Usually use this
Distance required to stop Use this formula
Prohibited Situations

Variable acceleration

Time explicitly required

Sign interpretation needed
Killer Trap

Formula gives

$$v^2$$

NOT

Direction of velocity.

Always analyse sign separately.

3.6 Formula 4

Average Velocity Form

$$s=\frac12(u+v)t$$

What It Does

Displacement equals:

Average Velocity × Time

Formula DNA

$$v_{avg}=\frac{u+v}{2}$$

Valid only under constant acceleration.

Eliminates

Acceleration (a)

Use When

Question Says Think
Initial velocity known Candidate
Final velocity known Candidate
Time known Candidate
Acceleration absent Strong Candidate
Biggest Trap

Many students use

$$v_{avg}=\frac{u+v}{2}$$

for every motion.

Wrong.

Valid only when acceleration is constant.

3.7 UAM Formula Selection Table

Trigger Seen Formula
Need velocity after time \(v=u+at\)
Need displacement after time \(s=ut+\frac12at^2\)
Time missing \(v^2=u^2+2as\)
Know u,v,t \(s=\frac12(u+v)t\)

3.8 Special UAM Results

Displacement During nth Second

Trigger:

nth Second Mentioned

Use the nth-second formula directly.

Average Velocity

Valid only when:

Acceleration Constant

Midpoint Velocity

Trigger:

Half Displacement

Use midpoint-displacement results.

Half-Time Velocity

Trigger:

Half Time
Common Confusion

Velocity at Half Time

is generally NOT equal to

Velocity at Half Displacement.

3.9 Top UAM Exam Traps

Trap 1

Negative Acceleration

≠

Slowing Down
Trap 2

Distance

≠

Displacement
Trap 3

Using $$v_{avg}=\frac{u+v}{2}$$ without constant acceleration.
Trap 4

Using UAM when acceleration varies.
Trap 5

Half Time

≠

Half Displacement
Trap 6

Retardation means acceleration opposite to velocity.

It does NOT necessarily mean negative acceleration.

UAM Quick Reaction Sheet

CONSTANT ACCELERATION?

NO
↓
Leave UAM

YES
↓
Time Missing?
↓
$$v^2=u^2+2as$$

Need Velocity?
↓
$$v=u+at$$

Need Displacement?
↓
$$s=ut+\frac12at^2$$

Know u,v,t?
↓
$$s=\frac12(u+v)t$$
Golden UAM Rule

Do not memorize four equations.

Memorize one idea:

Acceleration is Constant.

Everything else follows from it.

Continue Revision

Calculus Engine

Review differentiation and integration tools.

Sign Convention Engine

Master positive and negative directions.