Projectile Motion

Inquiry question: How can models that are used to explain projectile motion be used to analyse and make predictions?

Students:

• analyse the motion of projectiles by resolving the motion into horizontal and vertical components, making the following assumptions:
  • a constant vertical acceleration due to gravity
  • zero air resistance
• apply the modelling of projectile motion to quantitatively derive the relationships between the following variables:
  • initial velocity
  • launch angle
  • maximum height
  • time of flight
  • final velocity
  • launch height
  • horizontal range of the projectile (ACSPH099)

The Setup: Breaking Down Motion

Before deriving the specific variables, we must resolve the initial velocity vector ($u$) into its horizontal ($x$) and vertical ($y$) components.

Let:

• $u$ = Initial velocity magnitude
• $\theta$ = Launch angle
• $g$ = Acceleration due to gravity (acting downwards, so we use $-g$)
• $t$ = Time

Horizontal Component (Constant Velocity):

$$ u_x = u \cos \theta $$

$$ a_x = 0 $$

Vertical Component (Constant Acceleration):

$$ u_y = u \sin \theta $$

$$ a_y = -g $$

1. Time of Flight ($T$)

• Assumption: The projectile is launched from the ground and lands on the ground (Vertical displacement $\Delta y = 0$).*