Motion in a Plane - JEE Physics Notes

Introduction

Motion in a plane deals with two-dimensional motion, where an object moves in both the x and y directions simultaneously. This chapter is essential for JEE Physics as it covers vector representation, projectile motion, and circular motion, forming the foundation for advanced mechanics.

Vectors and Scalars

1. Scalars

Scalars are physical quantities that have only magnitude and no direction. Examples include:

• Mass
• Time
• Speed
• Temperature

2. Vectors

Vectors are quantities that have both magnitude and direction. Examples include:

• Displacement
• Velocity
• Acceleration
• Force

Vector Operations

1. Addition and Subtraction of Vectors

• Triangle Law: If two vectors are represented by the two sides of a triangle in order, their resultant is given by the third side in reverse order.
• Parallelogram Law: If two vectors act simultaneously at a point, their resultant is represented by the diagonal of the parallelogram formed by the vectors.

2. Resolution of Vectors

A vector can be resolved into two perpendicular components:

• Horizontal Component: A_x = A cos θ
• Vertical Component: A_y = A sin θ

Motion in Two Dimensions

1. Position and Displacement

• Position is represented by coordinates (x, y).
• Displacement is the shortest distance between the initial and final position.

2. Velocity in Two Dimensions

• Average Velocity: v_avg = Δr / Δt
• Instantaneous Velocity: Derivative of position with respect to time.

3. Acceleration in Two Dimensions

• Average Acceleration: a_avg = Δv / Δt
• Instantaneous Acceleration: Derivative of velocity with respect to time.

Projectile Motion

1. Equations of Projectile Motion

• Horizontal Range: R = (u² sin 2θ) / g
• Time of Flight: T = (2u sin θ) / g
• Maximum Height: H = (u² sin² θ) / 2g

2. Characteristics of Projectile Motion

• The horizontal motion is uniform (constant velocity).
• The vertical motion is uniformly accelerated (due to gravity).
• The path followed is a parabola.

Uniform Circular Motion

1. Definition

Uniform circular motion occurs when an object moves in a circular path with a constant speed but changing velocity due to direction change.

2. Angular Variables

• Angular Displacement: θ = s / r
• Angular Velocity: ω = dθ / dt
• Angular Acceleration: α = dω / dt

3. Centripetal Acceleration and Force

• Centripetal Acceleration: a_c = v² / r
• Centripetal Force: F_c = mv² / r

Relative Motion in Two Dimensions

1. Relative Velocity

The velocity of an object with respect to another object is given by:

• V_AB = V_A - V_B
• If objects move perpendicular to each other, the resultant velocity is found using the Pythagoras theorem.

Conclusion

Understanding Motion in a Plane is critical for solving problems related to vectors, projectile motion, and circular motion in JEE. Mastery of these concepts helps in tackling real-world physics problems and advanced mechanics efficiently.