# 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
• substituting  Viy = Vi × sin θ, max height can be given by
• 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
• relation
• so 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
• Final velocity can be calculated by
• Energy consideration (Conservation of mechanical energy)

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