Motion in a Plane – Class 11 Physics

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🌍 Introduction — Types of Motion

Motion is a change in position of an object with time. Motion can be of three types:

1D Motion

Rectilinear

Along a straight line — toy car pushed forward

2D Motion

In a Plane

Cricket ball hit for a sixer — projectile

3D Motion

In Space

Aeroplane flying through the sky

In rectilinear motion — force and velocity are along the same line
For 2D and 3D motion, we use vector quantities — they make equations elegant
This chapter focuses on motion in a plane — especially projectile motion and circular motion

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📏 Distance vs Displacement

These two are frequently confused — let us understand them clearly with a visual example.

📍 Path Length (Distance)

Total length of path actually travelled. Always positive. Scalar quantity.

d = total path covered

🎯 Displacement

Shortest straight-line distance from start to end point. Vector quantity — has direction.

s⃗ = x⃗₂ − x⃗₁

Key DifferenceIf object reverses direction → distance > |displacement| | displacement can be zero even if distance ≠ 0

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⚡ Velocity, Speed & Acceleration

1

Average Velocity

Displacement divided by time interval. Vector — same direction as displacement. v⃗ₐᵥ = Δx⃗ / Δt

2

Average Speed

Total path length divided by time. Scalar — always ≥ magnitude of average velocity.

3

Instantaneous Velocity

Velocity at a precise moment. Limiting value as Δt→0: v⃗ = dx⃗/dt — the derivative of position.

4

Acceleration

Rate of change of velocity. a⃗ = dv⃗/dt. Instantaneous acceleration = slope of tangent on v-t graph.

Always Remember — Uniform Motion Average velocity = Instantaneous velocity | Speed = |Velocity|

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📈 Position-Time & Velocity-Time Graphs

Graphs are the most powerful way to understand motion at a glance. Study these five cases carefully:

Horizontal line on x-t → object at rest (zero velocity)
Positive slope on x-t → uniform motion in +ve direction
Negative slope on x-t → uniform motion in −ve direction
Straight slope on v-t → uniform acceleration
Area under v-t graph = displacement of the object
Slope of v-t graph = acceleration (tangent for non-uniform)

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🔢 Equations of Motion (Uniform Acceleration)

Derived graphically from the velocity-time graph for uniform acceleration:

1st — from definition of a

v = u + at

velocity and time

2nd — area under v-t

s = ut + ½at²

displacement and time

3rd — eliminate t

v² = u² + 2as

no time needed

✦

Free Fall — most common example

Body starts from rest (u=0), falls under gravity alone. Acceleration = g = 9.8 m/s² downward. Distance ratio in successive equal intervals = 1 : 3 : 5 : 7 ...

Important — Derivation1st eq: a = (v−u)/t → v = u+at | 2nd eq: s = area of trapezium OABD = ut + ½at² | 3rd eq: eliminate t between 1st and 2nd

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🚂 Relative Velocity

When you are in a moving train and another train overtakes you, it appears to move slowly. That is relative motion!

Relative Velocity Formula v⃗ₐᵦ = v⃗ₐ − v⃗ᵦ (velocity of A with respect to B)
v⃗ᵦₐ = v⃗ᵦ − v⃗ₐ (velocity of B with respect to A)

|v⃗ₐᵦ| = |v⃗ᵦₐ| — magnitudes are equal, directions are opposite
Same direction: relative velocity = difference of speeds
Opposite directions: relative velocity = sum of speeds
Example: Plane A at 300 km/hr, Plane B at −350 km/hr → v_AB = 300−(−350) = 650 km/hr

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🧭 Motion in Two Dimensions — Vector Form

When motion is in a plane, we use x and y components separately. Position vector: r⃗ = x î + y ĵ

Key Equations — 2D Motion v⃗ₐᵥ = Δr⃗/Δt = [(x₂−x₁)î + (y₂−y₁)ĵ] / (t₂−t₁)

v⃗ = u⃗ + a⃗t s⃗ = u⃗t + ½a⃗t²

🌟 Most Important Principle Motion in 2D = Two independent rectilinear motions along x and y directions

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🎯 Projectile Motion

Any object thrown at an angle under gravity alone is a projectile. Its path is always a parabola!

↔ Horizontal (x)

No force → constant velocity

vₓ = u cosθ (constant)
sₓ = u cosθ · t

↕ Vertical (y)

Gravity acts → velocity changes

vᵧ = u sinθ − gt
sᵧ = u sinθ·t − ½gt²

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📐 Projectile — Key Formulas

T

Time of Flight

Total time in air = 2 × time to reach max height. At max height, vᵧ = 0.

R

Horizontal Range

Total horizontal distance covered. Maximum when θ = 45°.

H

Maximum Height

Highest point reached. Depends on vertical component of initial velocity.

🌟 Three Master Formulas T = 2u sinθ / g R = u² sin2θ / g H = u² sin²θ / 2g

Trajectory Equation — Parabola y = x tanθ − [g / (2u² cos²θ)] x² → form y = Ax + Bx² (parabola!)

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✏️ Solved Example — Projectile

Q: A stone is thrown with uₓ = 15 m/s, uᵧ = 20 m/s. Find velocity and position after 3s, max height and horizontal range. (g = 10 m/s²)

1

After 3 seconds:

vₓ = 15 m/s (unchanged) vᵧ = 20 − 10×3 = −10 m/s (downward)

2

Speed = √(15² + 10²) = √325 ≈ 18.03 m/s

Direction: tan α = 10/15 → α = 33°41' below horizontal

3

Position: sₓ = 15×3 = 45 m, sᵧ = 20×3 − 5×9 = 15 m

4

Max Height H = uᵧ²/2g = 400/20 = 20 m

5

Range R = 2·uₓ·uᵧ/g = 2×15×20/10 = 60 m

v = 18.03 m/s | H = 20 m | R = 60 m

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🔵 Uniform Circular Motion (UCM)

Object moves with constant speed along a circular path. Speed is constant but velocity direction always changes — so there IS acceleration!

Key Quantities Angular speed ω = 2π/T = v/r Period T = 2πr/v

🌟 Centripetal Acceleration & Force a = ω²r = v²/r (always directed towards centre)
F = mω²r = mv²/r Centripetal force

Acceleration is perpendicular to velocity — that is why speed stays constant
Moon orbiting Earth — gravity provides centripetal force
Car turning on a road — friction provides centripetal force

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🏁 Chapter Summary & Quick Revision

1

Distance vs Displacement

Distance = scalar path length. Displacement = vector, shortest path.

2

Equations of Motion

v=u+at, s=ut+½at², v²=u²+2as — for uniform acceleration only.

3

Relative Velocity

v⃗ₐᵦ = v⃗ₐ − v⃗ᵦ. Opposite directions → add magnitudes.

4

Projectile Motion

Horizontal: constant velocity. Vertical: uniform acceleration g. Path = parabola. R_max at θ=45°.

5

Uniform Circular Motion

Constant speed, changing direction. Centripetal a = v²/r towards centre. F = mv²/r.

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