Kinematics of Projectiles: Motion, Trajectory, and Formulas

Posted on Nov 6, 2025 in Physics

Kinematics of Projectile Motion

Defining Projectiles

  • Projectiles are bodies projected into the air that possess both horizontal and vertical components of motion.
  • Examples include: shot put, discus, javelin, and the human body during a jump.
  • Gravity determines the maximum height achieved by the projectile.
  • The horizontal component determines the maximum distance (range) the projectile reaches.
  • In real-world scenarios, only air and wind resistance significantly affect the projectile’s motion.

Note on Air Resistance: A projectile subject to air resistance drops sooner than one without.

Projectiles and the Effect of Gravity

Once an object is thrown into the air, gravity exerts a negative acceleration:

  • Acceleration due to gravity is -9.81 m/s2 (reducing upward velocity) until the apex (maximum height) is reached.
  • At the apex, the vertical velocity is zero.
  • The object’s velocity then accelerates downward at 9.81 m/s2 until reaching the ground.
  • If the object is caught at the same height it was released, the final velocity is equal in magnitude to the release velocity.

Trajectory: The Flight Path

Trajectory is defined as the flight path of a projectile. When neglecting air and wind resistance, the trajectory is influenced by three factors:

  1. Angle of Projection (resulting in a parabolic path)
  2. Initial Speed
  3. Relative Height of Release/Landing

Angle of Projection Principles

  • If the angle of projection is 90 degrees (e.g., straight up, like a dunk), the projectile will travel straight up and back down again.
  • If the angle is between 0 and 90 degrees (e.g., a soccer kick), the projectile will follow a parabolic path.
  • If the angle is 0 degrees (e.g., horizontal to the ground, like a baseball pitch), the projectile path will form half a parabola.

Optimum Angle for Maximum Displacement

  • When the projection and landing height are the same (e.g., a soccer kick), the optimum angle of projection for maximum horizontal displacement is 45°.
  • If the projection height is higher than the landing height, an angle less than 45° is necessary.
  • If the landing height is higher than the projection height, an angle greater than 45° is required.

Horizontal Component (Neglecting Wind Resistance)

The horizontal component always maintains a constant velocity and zero acceleration.

Key Principle: The horizontal component of a projectile has a constant velocity, while the vertical component changes due to gravity.

Projectile Kinematics Formulas

Formula 1: Final Velocity Calculation

v2 = v1 + a(t)

Example 1 (Vertical Velocity):

A golf ball is hit from a resting position and accelerates vertically at 4 m/s2 for 3 seconds until it stops accelerating. What is the vertical velocity of the golf ball at that moment?

Calculation:

v2 = 0 m/s + (4 m/s2)(3 s)

v2 = 12 m/s

The golf ball is traveling 12 m/s in a vertical direction when it stops accelerating.

Example 2 (General Acceleration):

What is the final velocity of an object that accelerates 3 m/s² for 8 seconds from an initial velocity of 12 m/s?

Calculation:

v2 = 12 + 3(8)

v2 = 36 m/s

Example 3 (Object Dropped):

If an object is dropped, the initial velocity (v1) is 0.

What is the velocity of an object dropped for 2.5 seconds?

Calculation (using a = 9.81 m/s2):

v2 = 0 + 9.81 (2.5)

v2 = 24.5 m/s

Formula 2: Displacement Calculation

d = v1(t) + 0.5a(t2)

Note: When calculating vertical displacement due to gravity, 0.5 × 9.81 m/s2 ≈ 4.9 m/s2.

Example 1 (Vertical Drop Distance):

A water balloon is dropped from a balcony and falls for 3 seconds before it hits the ground. How high is the balcony?

Calculation (using v1=0 and a=9.81 m/s2):

d = 0(3) + 0.5(9.81)(32)

d = 4.905(9)

d ≈ 44.1 m

The balcony is approximately 44.1 meters high.

Example 2 (Horizontal Distance):

How far will an object travel horizontally at 5 m/s for 3 seconds?

Calculation (Since horizontal acceleration is 0):

d = 5 (3)

d = 15 m

Example 3 (Vertical Fall Distance Check):

How far will an object fall after 3 seconds?

Calculation (using v1=0):

d = 0 (3) + ½ (9.81) (32)

d = 4.9 (9)

d = 44.1 m

Formula 3: Velocity-Displacement Relationship

v22 = v12 + 2ad

Note: There will be no question regarding Formula 3 on the final examination.