Projectile Motion (HL IB Physics)

What is a projectile?

• A projectile is a particle moving freely (non-powered), under gravity, in a two-dimensional plane
• Examples of projectile motion include throwing a ball, jumping off a diving board and hitting a baseball with a baseball bat
• In these examples, it is assumed that:
  • Resistance from the air or liquid (known as fluid resistance) the object is travelling through is negligible
  • Acceleration due to free-fall, g is constant as the object is moving close to the surface of the Earth

Examples of objects in a projectile motion trajectory

• An object is sent into a projectile motion trajectory with a resultant velocity, u at an angle, θ to the horizontal
  • Examples of this include a ball thrown from a height and a cannonball launched from a cannon

An object in a projectile motion trajectory has a resultant velocity at a given angle to the horizontal ground

• Some key terms to know, and how to calculate them, are:
  • Time of flight (total time): how long the projectile is in the air.
    • For typical projectile motion, the time to the maximum height is half of the total time
  • Maximum height attained: the height at which the projectile is momentarily at rest
    • This is when the vertical velocity component = 0
    • When the projectile is released and lands on the ground the projectile is at its maximum height when half of its total time has elapsed
  • Range: the horizontal distance travelled by the projectile

An object in projectile motion will have a vertical velocity of zero at maximum height when half the time has elapsed

Horizontal and Vertical Components

• The trajectory of an object undergoing projectile motion consists of a vertical component and a horizontal component
  • These quantities are independent of each other
  • Displacement, velocity and acceleration are all vector quantities that are different in both components
  • They need to be evaluated separately using the SUVAT Equations

	Horizontal Component	Vertical Component
Displacement	Maximum range at the end of the motion when the total time has elapsed Half the range at the maximum height when half the time has elapsed	Maximum height is at the top of the motion when half the time has elapsed
Velocity	Constant	Zero at maximum height
Acceleration	Zero (because velocity remains constant)	Acceleration of free fall, g = 9.8 ms−2   Positive when an object is falling towards Earth Negative when an object is moving away from Earth

Acceleration and horizontal velocity are always constant whilst vertical velocity changes

• The resultant velocity of an object in projectile motion can be split into its horizontal and vertical vector components using trigonometry where:
  • Vertical component = opposite side of the projectile triangle
    • opposite = sinθ × hyp = u sinθ
  • Horizontal component = adjacent side of the projectile triangle
    • adjacent = cosθ × hyp = u cosθ

The resultant velocity at an angle to the horizontal can be resolved using trigonometry into the horizontal and vertical components

• It can be helpful to see how different equations calculate different quantities using SUVAT equations
• Examples of obtaining the equations for total time, maximum height and range are shown below

Examples of using SUVAT equations to determine the time of flight, maximum height and range of a projectile

Solving problems with projectiles

• You may be required to calculate the missing quantities from the following projectile motion scenarios:
  • Vertical projection above the horizontal
  • Vertical projection below the horizontal
  • Horizontal projection
  • Projection at an angle, the most common scenario