Projectile Motion: Horizontal Launch

Introduction

• Discussion on the motion of a projectile fired horizontally from a height.
• Focus on calculating time to reach the ground and the angle at which it hits the ground.

Setup

• Consider a projectile fired horizontally from a point of origin at height H.
• Variables:
  • V₀ₓ: Initial horizontal velocity.
  • X, Y: Position coordinates of the projectile.
  • Assumptions: No initial vertical velocity, only horizontal.

Forces and Motion

• Gravity: Acts only in the vertical direction.
  • No horizontal force, thus horizontal velocity V₀ₓ remains constant.
• Vertical Motion:
  • Gravity causes an increase in vertical velocity (Vᵧ) as it falls.

Time to Reach the Ground

• Motion is considered along the y-axis.
• Key Equation:
  • H = 1/2 * g * T² where g is gravity, T is time to hit the ground.
  • Solving yields: T = sqrt(2H/g).

Horizontal Range

• Equation for horizontal motion:
  • S = V₀ₓ * T as there is no horizontal acceleration.
  • Horizontal range S = V₀ₓ * sqrt(2H/g).

Vertical Velocity at Any Time

• Equation: Vᵧ = g * T.
• Calculate vertical velocity if time T is known.*

Total Velocity

• Components:
  • Horizontal velocity (V₀ₓ) is constant.
  • Vertical velocity (Vᵧ) changes with time.
• Total Velocity (V_total) Equation:
  • V_total = sqrt(V₀ₓ² + Vᵧ²).

Angle of Impact

• Calculation of angle (θ):
  • Using triangle involving horizontal and vertical velocities.
  • tan(θ) = Vᵧ / V₀ₓ.
  • θ = tan⁻¹(Vᵧ / V₀ₓ).

Final Notes

• Emphasis on deriving equations from kinematics rather than memorizing.
• Understand the relationship between horizontal and vertical components of motion.