Projectile Motion: Equations, Formulas & Real-Life Examples
Master the parabolic path of objects thrown or launched under gravity. Learn all key equations, how to calculate range and maximum height, and see real-world applications in sports and daily life.
What Is Projectile Motion?
Projectile motion is the motion of an object that is thrown, launched, or projected into the air and then moves under the influence of gravity alone (ignoring air resistance).
The path followed by the projectile is a **parabola**. This is one of the most important applications of Newton’s laws of motion in classical mechanics.
Examples: A basketball shot, a cannonball, a jumping athlete, or a thrown ball.
Important Assumptions
- ✅ Air resistance is neglected
- ✅ Acceleration due to gravity (g = 9.8 m/s²) is constant and downward
- ✅ Horizontal velocity remains constant
- ✅ Vertical motion is uniformly accelerated
Components of Velocity
Initial velocity is split into two independent components:
- Horizontal (x-direction): vx = v × cosθ (constant)
- Vertical (y-direction): vy = v × sinθ (changes due to gravity)
Key Projectile Motion Equations
1. Time of Flight (T)
T = (2 × v × sinθ) / g
2. Maximum Height (H)
H = (v² × sin²θ) / (2g)
3. Horizontal Range (R)
R = (v² × sin2θ) / g
Maximum range occurs at θ = 45°
4. Equation of Trajectory
y = x tanθ – (g x²) / (2 v² cos²θ)
Real-Life Examples of Projectile Motion
Basketball Shot
The ball follows a parabolic path. Good shooters adjust angle and initial speed for maximum range and accuracy.
Football Kick / Soccer
Curved or high kicks are classic projectile motion with different launch angles.
Archery or Cannon Fire
Ancient cannons and modern artillery calculate range using these exact equations.
Long Jump / High Jump
Athletes optimize their launch angle for maximum horizontal distance.
Solved Practice Problems
1. A ball is kicked with initial velocity 20 m/s at an angle of 45°. Calculate the range (g = 10 m/s²).
Answer: R = (v² sin2θ)/g = (400 × 1)/10 = 40 meters
2. At what angle is the range maximum?
Answer: 45° (because sin2θ is maximum when 2θ = 90°)
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