Apply the modelling of projectile motion to quantitatively derive the relationships

Apply the modelling of projectile motion to quantitatively derive the relationships

Diagram of projectile motion :


In the diagram :

  • Hl : launch height
  • Vi : initial velocity
  • Vf : final velocity
  • θ: launch angle
  • H: maximum height
  • The distance AB is the ‘horizontal range of the projectile’ .
  • The time taken for projectile to travel along the trajectory from O to B is ‘time of flight’

Throughout the motion the vertical movement is affected  by gravitational acceleration  g = 9.8ms-2

To analyse the quantitative relationships between the variables :

  • The vertical and horizontal component of the initial velocity depends on the launch angle
    • Horizontal component : Vix = Vi × cos θ
    • Vertical component : Viy = Vi × sin θ
  • Relationship between max height, launch height, initial velocity and launch angle
    • max height is when vertical component of velocity is zero
    • Vy2 = Vy2 – 2g(H-Hl) where  is Vy vertical velocity and H is height
    • So when Vy=0, then max-height
    • substituting  Viy = Vi × sin θ, max height can be given by
      • max-height-2
  • Relationship between time of flight . initial velocity, angle of launch and launch height
    • time of flight is the time taken for the vertical component of the projectile to reach max height and reach the bottom.
    • using equation :


    • where displacement2-300x24-1
    • relationrelation
    • so time :time
  • Horizontal range of projectile
    • Since the horizontal component of projectile does not experience any acceleration , the horizontal velocity remains constant .
    • So horizontal range is simply Horizontal component of speed X time of flight
    • horizontal-range-300x26-1
    • horizontal-range-2-300x34-1
  • Final velocity can be calculated by
    • Energy consideration (Conservation of mechanical energy)
    • final-velocity-300x24-1

Extract from Physics Stage 6 Syllabus © 2017 NSW Education Standards Authority (NESA)