Chapter 3: Motion in a Plane

Chapter 3: Motion in a Plane

1. Scalars and Vectors

• Scalar Quantity: Has magnitude only (e.g., speed, mass)
• Vector Quantity: Has magnitude and direction (e.g., velocity, force)

2. Vector Basics

• Position Vector: Represents location of a point
• Displacement Vector: Change in position
• Equal vectors → same magnitude & direction
• Multiplication by scalar changes magnitude

3. Addition and Subtraction of Vectors

• Triangle Law of Addition
• Parallelogram Law of Addition
• Subtraction → adding negative vector

4. Unit Vector

A vector with magnitude 1.

i → x-direction
j → y-direction
k → z-direction

5. Resolution of Vectors

Breaking a vector into components.

A = Ax i + Ay j
Ax = A cosθ
Ay = A sinθ

6. Scalar (Dot) Product

A · B = AB cosθ

• Result is scalar
• Used in work done

7. Vector (Cross) Product

A × B = AB sinθ

• Result is vector
• Direction → right-hand rule

8. Motion in a Plane

Motion having two dimensions (x and y).

• Velocity has two components
• Analyzed independently in x and y directions

9. Projectile Motion

• Motion under gravity in 2D
• Horizontal velocity → constant
• Vertical motion → uniformly accelerated

Range = (u² sin2θ)/g
Time = (2u sinθ)/g
Max Height = (u² sin²θ)/2g

10. Uniform Circular Motion

• Speed constant but direction changes
• Acceleration toward center (centripetal)

a = v²/r

11. MCQ One-Liners

• Scalar has only magnitude.
• Vector has magnitude and direction.
• Unit vector magnitude is 1.
• i, j, k represent unit vectors.
• Dot product gives scalar.
• Cross product gives vector.
• Angle between perpendicular vectors = 90°.
• Dot product of perpendicular vectors = 0.
• Cross product of parallel vectors = 0.
• Projectile motion is 2D motion.
• Acceleration in projectile is g downward.
• Range is maximum at 45°.
• Circular motion has centripetal acceleration.
• Velocity direction changes in circular motion
• Horizontal velocity in projectile is constant.
• Vertical velocity changes due to gravity.
• Vector addition follows triangle law.
• Resolution splits vector into components.
• Magnitude of resultant depends on angle.
• Uniform circular motion has constant speed.

12. Numerical Problems with Solutions

1. Resolve Vector

A = 10 at 30°

Ax = 10 cos30° = 8.66
Ay = 10 sin30° = 5

2. Dot Product

A = 5, B = 4, θ = 60°

A·B = 5×4×cos60° = 20×0.5 = 10

3. Cross Product

A = 5, B = 4, θ = 90°

A×B = 5×4×sin90° = 20

4. Projectile Range

u = 20 m/s, θ = 30°, g = 10

Range = (u² sin2θ)/g
= (400 × sin60)/10
= (400 × 0.866)/10 = 34.64 m

5. Time of Flight

u = 20 m/s, θ = 30°

T = (2u sinθ)/g
= (40 × 0.5)/10 = 2 sec

6. Maximum Height

u = 20 m/s, θ = 30°

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

7. Centripetal Acceleration

v = 10 m/s, r = 5 m

a = v²/r = 100/5 = 20 m/s²

8. Resultant Vector

Two perpendicular vectors 3 and 4

R = √(3² + 4²) = √25 = 5