Motion in a Plane – Class 11 Physics NCERT | Projectile Motion, Vectors, Derivations & Formulas

🌍 Motion in a Plane – Class 11 Physics (NCERT)

Motion in a Plane deals with movement in two dimensions, using the concepts of vectors and projectile motion. This is the second part of kinematics after motion in a straight line.

📘 Key Concepts

• Scalars: Quantities with magnitude only (e.g., speed, mass)
• Vectors: Quantities with both magnitude and direction (e.g., velocity, displacement)
• Vector Addition: Using triangle or parallelogram law
• Resolution of Vectors: Splitting a vector into components along x and y axes

🧮 Basic Vector Formulas

• Resultant of two vectors:
  R = √(A² + B² + 2ABcosθ)
• Component form:
  A = A_xî + A_yĵ
• Magnitude:
  |A| = √(A_x² + A_y²)
• Direction:
  θ = tan⁻¹(A_y/A_x)

🎯 Projectile Motion

When an object is thrown at an angle, it follows a curved path called a projectile. It is a 2D motion with horizontal and vertical components.

📌 Assumptions:

• Acceleration due to gravity acts vertically downward
• Horizontal velocity remains constant
• Air resistance is neglected

🧾 Equations for Projectile Motion

Quantity	Formula
Time of flight (T)	T = (2u sinθ)/g
Maximum height (H)	H = (u² sin²θ)/(2g)
Horizontal range (R)	R = (u² sin2θ)/g

Component-wise Equations

Let initial velocity be u at angle θ with horizontal:

• Horizontal velocity: u_x = u cosθ
• Vertical velocity: u_y = u sinθ
• Horizontal displacement: x = u cosθ × t
• Vertical displacement: y = u sinθ × t – ½gt²

🔄 Circular Motion (Brief)

Circular motion is also motion in a plane, where the object moves in a circle of radius r.

• Velocity direction changes continuously
• Centripetal acceleration:
  a_c = v²/r
• Centripetal force:
  F = mv²/r

📋 Summary Table

Concept	Formula
Resultant Vector	R = √(A² + B² + 2ABcosθ)
Time of Flight	T = (2u sinθ)/g
Max Height	H = (u² sin²θ)/(2g)
Range	R = (u² sin2θ)/g
Centripetal Acceleration	a = v²/r

📝 Final Notes

• Always resolve motion into x and y components
• Use vector algebra for 2D analysis
• Projectile motion is symmetrical (if on level ground)

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