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# 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 :
*Vi*_{x}= Vi × cos θ - Vertical component :
*Vi*_{y}= Vi × sin θ

- Horizontal component :

- Relationship between max height, launch height, initial velocity and launch angle
- max height is when vertical component of velocity is zero
*V*where is_{y}^{2}= V_{y}^{2}– 2g(H-Hl)*V*vertical velocity and_{y}*H*is height- So when
*V*, then_{y}=0

- substituting
*Vi*, max height can be given by_{y}= V_{i}× sin θ

- substituting

- 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)