Understanding Projectile Motion

Think back to the last time you threw a ball, kicked a soccer ball, or watched fireworks shoot into the sky.
That curved path you saw — going up, slowing down, reaching its peak, and then coming down — is called projectile motion.

It’s one of the most fundamental topics in physics, because it combines both horizontal motion and vertical motion under the influence of gravity.

This calculator helps you solve projectile motion problems instantly, without juggling multiple equations on paper.

Why Is It Important?

Projectile motion isn’t just a classroom exercise. It’s applied in:

• 🎮 Game design (simulating bullets, arrows, or thrown objects).
  
• ⚽ Sports science (measuring how far a ball can travel).
  
• 🏗️ Engineering (trajectory of machines, construction safety).
  
• 🚀 Space & defense (rockets, missiles, satellites).
  

In short — if something moves through the air, projectile motion equations are behind it.

The Core Formulas

Projectile motion is based on physics equations that combine velocity, angle, and gravity:

1. Time of Flight (T):

   \[ T = \frac{2 u \cdot \sin(\theta)}{g} \]

   (where \(u\) = initial velocity, \(\theta\) = launch angle, \(g\) = gravity)

2. Maximum Height (H):

   \[ H = \frac{u^2 \cdot \sin^2(\theta)}{2 g} \]

3. Range (R):

   \[ R = \frac{u^2 \cdot \sin(2\theta)}{g} \]

4. Final Velocity (V):

   Depends on both horizontal and vertical components at impact:

   \[ V = \sqrt{V_x^2 + V_y^2} \]

How the Calculator Works (Step-by-Step)

1. Enter Initial Velocity (m/s):
   How fast the object is thrown/launched.
   
2. Enter Launch Angle (degrees):
   Angle above the ground (0° = flat, 90° = straight up).
   
3. Enter Initial Height (optional):
   Useful if launching from a raised platform, building, or cliff.
   
4. Gravity (default 9.81 m/s²):
   Earth’s standard gravity. You can change it for other planets.
   
5. Click Calculate.
   The tool instantly gives you:
   
   • ⏱ Time of Flight (how long it stays in the air)
     
   • 📈 Maximum Height (the peak point)
     
   • 📏 Range (how far it travels horizontally)
     
   • 🚀 Final Velocity (speed just before hitting the ground)
     

Example

• Velocity: 20 m/s
  
• Angle: 45°
  
• Height: 0 m
  

Results:

• Time of Flight ≈ 2.9 s
  
• Max Height ≈ 10.2 m
  
• Range ≈ 40.8 m
  
• Final Velocity ≈ 20 m/s (back to same speed, different direction)
  

This shows how the motion is perfectly symmetrical when launched from ground level.

FAQ