Projectile Motion

Introduction

• Projectile Motion is a type of motion under the influence of gravity.
• Involves motion in a plane (2D motion), which can be analyzed as motion in x (horizontal) and y (vertical) directions.
• Examples include a ball thrown and trajectories like football kicks.

Fundamental Concepts

• Plane Motion: Involves both x and y-direction components, unlike straight-line motion which is 1D.
• Components of Initial Speed:
  • u_x = u cos(θ): Initial speed in x-direction.
  • u_y = u sin(θ): Initial speed in y-direction.
• Acceleration:
  • a_x = 0: No acceleration in the x-direction.
  • a_y = -g: Gravity in the y-direction, always acting downwards.
• Constant Velocity in x and changing velocity in y due to gravity.

Time of Flight (T)

• Formula: T = 2u sin(θ) / g
• Derivation:
  1. Consider motion in Y direction and point A to B.
  2. Initial Velocity u_y = u sin(θ), Final Velocity v_y = 0
  3. Apply v = u + at
  4. Final Velocity = 0: 0 = u sin(θ) - gt
  5. t = u sin(θ) / g
  6. Total Time of Flight T = 2t
• Verification with entire motion consideration.

Maximum Height (H)

• Formula: H = (u^2 sin^2(θ)) / (2g)
• Derivation:
  1. Consider motion from point A to B in Y direction.
  2. Initial Velocity u_y = u sin(θ), Final Velocity v_y = 0
  3. Apply v² = u² + 2as to solve for displacement H.

Horizontal Range (R)

• Formula: R = (u^2 sin(2θ)) / g
• Also expressed as R = u^2 (2 sin(θ) cos(θ)) / g
• Derivation:
  1. Consider motion in X direction from point A to C.
  2. Initial Velocity u_x = u cos(θ), No X-direction Acceleration.
  3. Time T = 2u sin(θ) / g
  4. Horizontal Displacement s = ut + 0.5at² simplify to R = u² sin(2θ) / g

Velocity Components

• Initial Velocity: u_x = 4i, u_y = 3j
• Final Velocity at different points considering presence/absence of gravitational effect.
• Determining components after a specified time.

Miscellaneous Applications

• Deriving velocities and angles at different time intervals.
• Calculating displacement vectors.

Important Properties

• Range is the same for two different angles θ and 90-θ when U is constant.
  • Proof through Range formula and angle substitutions.

Special Cases

1. If Maximum Height H and Range R are equal.

• Derivatives and solving for angles.

2. Examine situations involving different initial projections and solving for max angles and ranges.

Equation of Trajectory

• Path in a projectile motion is parabolic.
• Derivation:
  1. Combine motions in X and Y directions.
  2. Eliminate time factor and integrate Y and X components.
  3. y = x tan(θ) - (g x²) / (2u² cos²(θ))
  • Relates x and y components in projectile motion.
• Questions might involve comparisons to polynomials and evaluating angles, initial velocity, etc.

Practical Problems

• Specific problems involving calculations of height, range, and angles from given components.
• Derivations involving vectors, Path distinction.

Conclusion

• Understanding of basic projectile motion with formulas for Time of Flight, Maximum Height, and Range.
• Utilization of equations in problem-solving related to angles, initial velocities, and displacements.
• Understanding projectile motion involves analyzing vector components and accelerations due to gravity.