LESSON–3 : MOTION IN A PLANE

ADDITIONAL MATERIALS – 3
IIT JEE PREVIOUS YEAR QUESTIONS & ANSWERS

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🔹 PYQ–1 (JEE MAIN)

Question:
A projectile is projected with velocity 20 m/s at an angle of 30°. Find its maximum height. (g = 10 m/s²)

Answer:

H = (u² sin²θ) / (2g)
= (400 × 0.25) / 20
= 5 m

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🔹 PYQ–2 (JEE MAIN)

Question:
Find the magnitude of vector A = 3î + 4ĵ.

Answer:

|A| = √(3² + 4²) = 5

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🔹 PYQ–3 (JEE MAIN)

Question:
Two particles move with velocities 10î + 2ĵ and 4î − 2ĵ. Find the relative velocity of first particle with respect to second.

Answer:

v_AB = (10 − 4)î + (2 + 2)ĵ
= 6î + 4ĵ

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🔹 PYQ–4 (JEE ADVANCED)

Question:
At the highest point of a projectile, which of the following is zero?
(a) Velocity
(b) Acceleration
(c) Vertical component of velocity
(d) Horizontal component of velocity

Answer:

Correct option: (c) Vertical component of velocity

Acceleration is not zero; it is always g downward.

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🔹 PYQ–5 (JEE ADVANCED)

Question:
A projectile is projected at angle θ with horizontal. At what angle will its range be maximum?

Answer:

Range R = (u² sin2θ)/g
Maximum when sin2θ = 1
⇒ 2θ = 90°
⇒ θ = 45°

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🔹 PYQ–6 (JEE ADVANCED – CONCEPTUAL)

Question:
Can the velocity of a projectile ever be perpendicular to its acceleration?

Answer:

Yes. At the highest point, velocity is horizontal and acceleration is vertical.

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🔹 PYQ–7 (JEE MAIN)

Question:
Two projectiles are thrown with same speed at angles 30° and 60°. Compare their ranges.

Answer:

Ranges are equal because:
sin60° = sin120°

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🔹 PYQ–8 (JEE ADVANCED)

Question:
A projectile is projected horizontally from the top of a tower. Which quantity remains constant during motion?

Answer:

Horizontal component of velocity remains constant.

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🔹 PYQ–9 (JEE MAIN)

Question:
A particle moves with velocity v = 2î + 3ĵ. Find its speed.

Answer:

Speed = √(2² + 3²) = √13 m/s

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🔹 PYQ–10 (JEE ADVANCED – THINKING)

Question:
Two bodies are projected simultaneously from the same point, one vertically upward and the other horizontally with same speed. Which one hits the ground first?

Answer:

Both hit the ground at the same time, because time depends only on vertical motion.

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🎯 PYQ SOLVING STRATEGY

• Always draw a rough diagram first
• Resolve velocity into components
• Use symmetry and complementary angle logic
• Do not memorise blindly – understand motion

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✅ ADDITIONAL MATERIALS – 3 (IIT PYQs) COMPLETED

 

LESSON–3 : MOTION IN A PLANE

ADDITIONAL MATERIALS – 2
ADVANCED PROBLEMS + THINKING TRICKS

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🔹 PART–A : ADVANCED PRACTICE PROBLEMS (10)

Problem 1: A vector has components 5î − 12ĵ. Find its magnitude.

Answer:

|A| = √(5² + 12²) = √169 = 13

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Problem 2: Find the angle made by vector 3î + √3 ĵ with x-axis.

Answer:

tanθ = (√3) / 3
θ = 30°

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Problem 3: A projectile is thrown with velocity 20 m/s at 60°. Find its time of flight. (g = 10 m/s²)

Answer:

T = (2u sinθ)/g
= (2 × 20 × sin60°)/10
= 3.46 s

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Problem 4: Find the maximum height reached by the projectile in Problem–3.

Answer:

H = (u² sin²θ)/(2g)
= (400 × 3/4) / 20 = 15 m

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Problem 5: Two projectiles are projected with same speed. One at 30° and the other at 60°. Which one stays longer in air?

Answer:

Time of flight ∝ sinθ
Since sin60° > sin30°, 60° projectile stays longer.

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Problem 6: A body is projected horizontally from a tower of height 45 m. Find the time taken to reach the ground. (g = 10 m/s²)

Answer:

h = ½gt²
45 = 5t²
t = 3 s

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Problem 7: Find the horizontal distance covered in Problem–6 if horizontal velocity is 10 m/s.

Answer:

Distance = u × t = 10 × 3 = 30 m

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Problem 8: Two cars move perpendicular to each other with speeds 20 m/s and 15 m/s. Find the magnitude of relative velocity.

Answer:

Relative velocity = √(20² + 15²)
= √625 = 25 m/s

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Problem 9: At what angle should a projectile be thrown to get maximum range?

Answer:

45°

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Problem 10: What is the acceleration of a projectile at the highest point?

Answer:

Acceleration = g downward
It is not zero.

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🔹 PART–B : ADVANCED THINKING TRICKS

🎯 Trick 1: Time of Flight Depends Only on Vertical Motion

Horizontal velocity does not affect time of flight. Only vertical component decides it.

🎯 Trick 2: Use sinθ Logic Instead of Formula

If speed is same:

• Higher angle → more time in air
• Lower angle → less time in air

🎯 Trick 3: Complementary Angle Shortcut

Angles θ and (90° − θ):

• Same range
• Different time of flight

🎯 Trick 4: Perpendicular Velocities

If velocities are perpendicular:

Resultant = √(v₁² + v₂²)

🎯 Trick 5: Highest Point Rule

At highest point:

• Vertical velocity = 0
• Horizontal velocity ≠ 0
• Acceleration always acts downward

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🏆 FINAL PRACTICE ADVICE

If you solve **Additional–1 + Additional–2** fully, you will be able to handle any Intermediate-level problem and many IIT-level basic problems from this chapter.

 

LESSON–3 : MOTION IN A PLANE

ADDITIONAL MATERIALS
IMPORTANT PROBLEMS + SMART TRICKS

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🔹 PART–A : IMPORTANT PROBLEMS (10)

Problem 1: Find the magnitude of a vector having components 6î + 8ĵ.

Answer:

|A| = √(6² + 8²) = √100 = 10

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Problem 2: A vector of magnitude 20 m/s makes an angle of 30° with horizontal. Find its horizontal component.

Answer:

v_x = v cosθ = 20 × cos30° = 17.32 m/s

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Problem 3: Find the vertical component of velocity 25 m/s at 60°.

Answer:

v_y = v sinθ = 25 × sin60° = 21.65 m/s

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Problem 4: A projectile is thrown with speed 10 m/s at 45°. Find its time of flight. (g = 10 m/s²)

Answer:

T = (2u sinθ)/g
= (2 × 10 × sin45°)/10 = 1.41 s

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Problem 5: Find the maximum height of a projectile thrown at 30° with speed 20 m/s.

Answer:

H = (u² sin²θ)/(2g)
= (400 × 0.25)/20 = 5 m

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Problem 6: Find the range of a projectile thrown at 45° with speed 20 m/s.

Answer:

R = (u² sin2θ)/g
= (400 × 1)/10 = 40 m

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Problem 7: Two projectiles are thrown with same speed at angles 30° and 60°. Compare their ranges.

Answer:

Ranges are equal because sin60° = sin120°.

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Problem 8: A body is projected horizontally with velocity 10 m/s. Find the horizontal distance travelled in 3 s.

Answer:

Distance = u × t = 10 × 3 = 30 m

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Problem 9: Two trains move in opposite directions with speeds 40 km/h and 60 km/h. Find relative velocity.

Answer:

Relative velocity = 40 + 60 = 100 km/h

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Problem 10: At the highest point of projectile motion, what is the velocity?

Answer:

Only horizontal component exists. Vertical component is zero.

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🔹 PART–B : SMART TRICKS & SHORTCUTS

🎯 Trick 1: Resolve First, Think Later

In any vector or projectile problem, first resolve velocity into horizontal and vertical components.

🎯 Trick 2: Remember Three Golden Formulas

• T = (2u sinθ)/g
• H = (u² sin²θ)/(2g)
• R = (u² sin2θ)/g

🎯 Trick 3: Complementary Angles

Projectiles fired at θ and (90° − θ) have equal range.

🎯 Trick 4: Horizontal Projection Rule

Horizontal motion is uniform. Vertical motion is free fall.

🎯 Trick 5: Relative Velocity Shortcut

• Same direction → subtract speeds
• Opposite direction → add speeds

🎯 Trick 6: Highest Point Logic

At highest point:

• Vertical velocity = 0
• Acceleration ≠ 0 (still g)

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🏆 FINAL LEARNING TIP

If you master these 10 problems and tricks, Motion in a Plane becomes a scoring chapter in both Intermediate and IIT exams.

 

IIT PHYSICS – LESSON 3

MOTION IN A PLANE
STAGE–4 : IIT PYQs + MIND MAP + HOW TO STUDY

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 Why Stage–4 Is Crucial for IIT Rank

Most students stop after learning formulas. IIT rankers go one step further — they master:

• Previous Year Questions (PYQs)
• Concept connections
• Speed + accuracy strategy

IIT Reality: If you can solve PYQs confidently, you are already above average.

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🔹 1️⃣ IIT JEE PREVIOUS YEAR QUESTIONS (SOLVED)

PYQ–1 (JEE Main)

Question:
A particle is projected with speed 10 m/s at an angle of 30° to the horizontal. Find its horizontal range. (g = 10 m/s²)

Solution:

Range, R = (u² sin2θ) / g
= (100 × sin60°) / 10
= (100 × √3/2) / 10
= 5√3 m

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PYQ–2 (JEE Main)

Question:
Find the magnitude of the vector A = 3î − 4ĵ.

Solution:

|A| = √(3² + 4²) = 5

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PYQ–3 (JEE Advanced)

Question:
Two projectiles are projected with same speed at angles θ and (90° − θ). Compare their times of flight.

Solution:

Time of flight, T = (2u sinθ) / g
For second projectile: T = (2u cosθ) / g

Conclusion:
Times of flight are different unless θ = 45°.

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PYQ–4 (JEE Advanced – Conceptual)

Question:
At the highest point of a projectile, is acceleration zero?

Answer:

No. Acceleration is always equal to g and acts vertically downward, even at the highest point.

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 2️⃣ CONCEPT MIND MAP (FAST REVISION)

• Vectors
  • Magnitude & direction
  • Resolution into components
• Velocity in a Plane
  • v = √(v_x² + v_y²)
  • Direction from tanθ = v_y/v_x
• Projectile Motion
  • Independent x and y motion
  • Parabolic trajectory
  • Time of flight, range, height
• Relative Velocity
  • v_AB = v_A − v_B
  • Vector subtraction

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📘 3️⃣ HOW TO STUDY THIS CHAPTER (IIT METHOD)

Phase	What to Do	Purpose
Concept	Master vector basics & diagrams	Avoid confusion
Practice	Solve 20–30 projectile problems	Speed building
PYQs	Solve last 15 years PYQs	Pattern recognition
Revision	Mind map + formula sheet	Long-term retention

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4️⃣ COMMON IIT MISTAKES

• Ignoring vector direction
• Mixing up sinθ and cosθ
• Forgetting g acts downward always
• Skipping diagram while solving problems

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 FINAL MESSAGE

If vectors and projectile motion become natural to you, Mechanics becomes your strength.

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NEXT: STAGE–5 – NECESSARY DIAGRAMS (WITH PLACEHOLDERS)

Intermediate Physics Complete Notes – IIT JEE & Board Exams

This page contains complete Intermediate Physics notes along with IIT JEE materials, previous year questions, solved problems, diagrams, and exam strategies.

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📘 Lesson 1: Units and Measurements

• Complete Notes
• Important Questions

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📘 Lesson 2: Motion in a Straight Line

• Complete Notes
• Important Questions
• IIT JEE Material
• IIT JEE PYQs & Strategy
• Diagrams

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📘 Lesson 3: Motion in a Plane

• Part 1
• Part 2
• Part 3
• Part 4
• Part 5
• Part 6
• Part 7
• Part 8
• Part 9
• Part 10
• Part 11
• Part 12
• Part 13
• Part 14
• Part 15
• Part 16
• Part 17

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📌 This page is updated regularly with new lessons.

 

IIT PHYSICS – LESSON 3

MOTION IN A PLANE
STAGE–3 : JEE MAIN & JEE ADVANCED MATERIAL

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🌟 Why Motion in a Plane is Important for IIT

This chapter introduces vector mechanics, which is the backbone of entire Mechanics. IIT questions test not memory, but vector reasoning and geometry.

IIT Insight: If vectors are clear, half of Mechanics becomes easy.

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1️⃣ Vectors – Mathematical Treatment

A vector is represented by an arrow. Its magnitude is the length of the arrow and direction is given by its orientation.

In Cartesian form:

Vector A = A_x î + A_y ĵ

Magnitude of vector:

|A| = √(A_x² + A_y²)

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2️⃣ Resolution of Vectors

Any vector can be resolved into perpendicular components.

If a vector A makes an angle θ with x-axis:

• A_x = A cosθ
• A_y = A sinθ

JEE Trick: Always draw a right-angled triangle before resolving vectors.

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3️⃣ Projectile Motion (IIT Perspective)

Projectile motion is two-dimensional motion under constant gravity.

• Horizontal acceleration = 0
• Vertical acceleration = g (downward)

Equations of motion:

• x = u cosθ · t
• y = u sinθ · t − ½gt²

Trajectory equation:

y = x tanθ − (g x²) / (2u² cos²θ)

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4️⃣ Important Results (Very High Weightage)

• Time of flight, T = (2u sinθ) / g
• Maximum height, H = (u² sin²θ) / (2g)
• Range, R = (u² sin2θ) / g

IIT Result: Maximum range occurs at θ = 45°

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5️⃣ Relative Velocity (Vector Approach)

Relative velocity of A with respect to B:

v_AB = v_A − v_B

Relative velocity problems are solved using:

• Vector subtraction
• Triangle law of vectors
• Parallelogram law

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6️⃣ JEE MAIN Level Questions

Q1: A projectile is thrown with velocity 20 m/s at 30°. Find its time of flight.

Solution:

T = (2u sinθ)/g
= (2 × 20 × 0.5) / 10 = 2 s

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Q2: Find the magnitude of vector having components (3î + 4ĵ).

Answer:

|A| = √(3² + 4²) = 5

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7️⃣ JEE ADVANCED Thinking Problem

Q: Two particles are projected simultaneously from same point with same speed but at angles θ and (90° − θ). Compare their ranges.

Answer:

Ranges are equal, because sin2θ = sin[2(90° − θ)].

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⚠️ Common IIT Mistakes

• Forgetting vector direction
• Wrong resolution of components
• Using scalar formulas for vector quantities

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➡️ NEXT: STAGE–4 – IIT PYQs + HOW TO STUDY + MIND MAP

 

INTERMEDIATE PHYSICS – LESSON 3

MOTION IN A PLANE
STAGE–2 : IMPORTANT QUESTIONS & NUMERICALS

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🔹 A. VERY SHORT ANSWER QUESTIONS (2 MARKS)

Q1. What is motion in a plane?

Motion in which a body moves in two dimensions simultaneously is called motion in a plane.

Q2. Define scalar quantity.

A physical quantity which has only magnitude and no direction is called a scalar quantity.

Q3. Define vector quantity.

A physical quantity which has both magnitude and direction is called a vector quantity.

Q4. What is a position vector?

The vector drawn from the origin to the position of a particle is called the position vector.

Q5. What is projectile motion?

The motion of a body projected into air and moving under the influence of gravity alone is called projectile motion.

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🔹 B. SHORT ANSWER QUESTIONS (4 MARKS)

Q1. Distinguish between scalar and vector quantities.

Scalar	Vector
Has only magnitude	Has magnitude and direction
Added algebraically	Added vectorially
Example: speed	Example: velocity

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Q2. Explain position vector and displacement vector.

The position vector represents the position of a particle with respect to the origin. The displacement vector represents the change in position of the particle.

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Q3. What is relative velocity?

The velocity of one body with respect to another body is called relative velocity.

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🔹 C. LONG ANSWER QUESTIONS (8 MARKS)

Q1. Explain projectile motion.

Projectile motion is the motion of a body projected into air and moving under gravity.

• The horizontal motion is uniform
• The vertical motion is uniformly accelerated
• The path followed is parabolic

Horizontal and vertical motions are independent of each other.

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Q2. Derive expressions for horizontal and vertical displacement of a projectile.

Horizontal displacement:

x = u cosθ · t

Vertical displacement:

y = u sinθ · t − ½gt²

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🔹 D. NUMERICAL PROBLEMS (VERY IMPORTANT)

Numerical 1: A projectile is thrown horizontally with a velocity of 10 m/s. Find the horizontal distance travelled in 2 s.

Solution:

Horizontal distance = u × t
= 10 × 2 = 20 m

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Numerical 2: Find the horizontal and vertical components of velocity 20 m/s making an angle of 30° with horizontal.

Solution:

v_x = v cosθ = 20 × cos30° = 17.32 m/s
v_y = v sinθ = 20 × sin30° = 10 m/s

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Numerical 3: Two trains move in the same direction with speeds 40 km/h and 60 km/h. Find the relative velocity.

Solution:

Relative velocity = 60 − 40 = 20 km/h

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🎯 INTERMEDIATE EXAM TIPS

• Always draw neat diagrams for projectile motion
• Write formulas clearly before substitution
• Vector questions are scoring if diagram is drawn
• Numericals are easy marks – practice daily

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➡️ NEXT: STAGE–3 – IIT MATERIAL (JEE MAIN + ADVANCED)

 

INTERMEDIATE PHYSICS – LESSON 3

MOTION IN A PLANE

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1. Introduction

Motion in a plane is a type of motion in which a particle moves in two dimensions simultaneously. Unlike motion in a straight line, here both x and y directions are involved.

Examples:

• Motion of a ball thrown into air
• Motion of a bird flying
• Motion of a stone projected at an angle

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2. Scalars and Vectors

(a) Scalar Quantities

Scalars have only magnitude and no direction.

Examples: distance, speed, mass, time

(b) Vector Quantities

Vectors have both magnitude and direction.

Examples: displacement, velocity, acceleration, force

                                

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3. Position and Displacement Vectors

The position of a particle in a plane is represented by a position vector drawn from origin to the point.

Displacement vector is the change in position vector.

Important: Displacement depends only on initial and final positions, not on the path.

📐 Insert Diagram: Position vector and displacement vector in XY-plane

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4. Velocity and Acceleration in a Plane

In two-dimensional motion, velocity has two components:

• Horizontal component (v_x)
• Vertical component (v_y)

Resultant velocity is obtained by vector addition.

Key Formula:
v = √(v_x² + v_y²)

📐 Insert Diagram: Horizontal and vertical components of velocity

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5. Projectile Motion

Projectile motion is the motion of a body thrown into air and moving under the influence of gravity alone.

Examples:

• A ball thrown at an angle
• A bullet fired from a gun

Assumptions:

• Air resistance is neglected
• Acceleration due to gravity is constant and acts downward

📐 Insert Diagram: Projectile motion showing parabolic path

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6. Horizontal Projection

When a body is projected horizontally from a height, it follows a curved (parabolic) path.

Horizontal motion is uniform, while vertical motion is uniformly accelerated.

Important Concept:
Horizontal and vertical motions are independent of each other.

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7. Relative Velocity (Basic Idea)

Relative velocity is the velocity of one object with respect to another.

Example: If two trains move in the same direction, relative velocity is the difference of their velocities.

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8. Important Formulae (Intermediate Level)

• Resultant velocity = √(v_x² + v_y²)
• Horizontal displacement = v_x t
• Vertical displacement = v_y t − ½gt²

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9. Board Exam Focus Points

• Difference between scalar and vector (2M / 4M)
• Projectile motion explanation (8M)
• Diagram-based questions are common
• Formula-based numericals are scoring

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➡️ NEXT: STAGE–2 – INTERMEDIATE EXAM QUESTIONS & NUMERICALS

Intermediate Physics Complete Notes – IIT JEE & Board Exams

This page contains complete Intermediate Physics notes along with IIT JEE materials, previous year questions, solved problems, diagrams, and exam strategies.

---

📘 Lesson 1: Units and Measurements

• Complete Notes
• Important Questions

---

📘 Lesson 2: Motion in a Straight Line

• Complete Notes
• Important Questions
• IIT JEE Material
• IIT JEE PYQs & Strategy
• Diagrams

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📘 Lesson 3: Motion in a Plane

• Part 1
• Part 2
• Part 3
• Part 4
• Part 5
• Part 6
• Part 7
• Part 8
• Part 9
• Part 10
• Part 11
• Part 12
• Part 13
• Part 14
• Part 15
• Part 16
• Part 17

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This page is updated regularly with new lessons.