Motion in Two Dimensions

Introduction

• Important for JEE aspirants.
• Previous series covered motion in one dimension.
• Current lecture focuses on motion in two dimensions with 10 numerical examples.

Types of Motion in Two Dimensions

1. Direct 2D Motion
   • Particle moves in a 2D plane (XY or YZ).
2. Projectile Motion
   • Special case where a particle is projected at an angle (theta) and follows a parabolic path.

Projectile Motion Types

• Ground to ground projectile.
• Projectile from a tower/building.
• Projectile on an inclined plane (up and down).
• Projectile from a moving trolley or car.

Key Concepts of Motion in Two Dimensions

• Representation in XY-plane.
• Equation of trajectory:
  • If a particle's path is represented as ( y = f(x) ), it's the equation of trajectory.

Velocity Components

• Velocity has two components:
  • ( v_x ) and ( v_y ).
• 2D motion is a vector sum of independent 1D motions.

Motion Equations

• For uniform acceleration:
  • ( v = u + at ) (both x and y directions)
  • ( s = ut + \frac{1}{2} at^2 ) (used separately in each direction)

Position Vector

• Position vector ( \vec{r} = x \hat{i} + y \hat{j} )
• Determine ( x ) and ( y ) from kinematic equations.

Example Problem

• A particle starts from the origin with:
  • Initial velocity ( u = 10 , m/s ) (in y-direction)
  • Constant acceleration: ( \vec{a} = 8\hat{i} + 2\hat{j} , m/s^2 )
• Question:
  • Find time when ( x = 16 , m ), ( y ) coordinate, and velocity at that time.

Steps to Solve

1. Find time using x-motion:
   • ( 16 = 0 + \frac{1}{2}(8)(t^2) ) ➔ ( t = 2 , s )
2. Use y-motion to find y-coordinate:
   • ( y = 10t + \frac{1}{2}(2)(t^2) ) ➔ ( y = 24 , m )
3. Find velocities:
   • ( v_x = u_x + a_x t = 16 , m/s )
   • ( v_y = u_y + a_y t = 14 , m/s )
   • Resultant velocity ( v = \sqrt{v_x^2 + v_y^2} \approx 21.3 , m/s )

Projectile Motion

• Particle projected at an angle ( \theta ) follows parabolic path:
• Important points:
  • Height from which it was projected matters.
• Standard formulas:
  • Time of flight: ( T = \frac{2u_y}{g} )
  • Maximum height: ( H_{max} = \frac{u_y^2}{2g} )
  • Range: ( R = \frac{u_x u_y}{g} )

Conditions for Maximum Range

• For constant speed, angle of projection for max range is 45°.
• For complementary angles, the ranges are equal but heights can differ.

Equations at Time ( t )

• ( x = u_x t )
• ( y = u_y t - \frac{1}{2} gt^2 )

Examples and Applications

1. Projectile from Moving Trolley

   • Combined effects of horizontal motion from the trolley.
   • Must consider relative speeds when calculating trajectories.

2. Dropping from a Helicopter

   • Use same principles to find what distance to drop a package relative to a victim on the ground.

3. Additional Practice

   • Included various mechanisms for solving projectile motion problems and tracking motions in inclined planes and depending on heights.

Conclusion

• Fundamental understanding of projectile motion and motion in two dimensions is crucial for solving JEE questions.
• Review relevant equations and practice numerical to strengthen concepts.