Projectile Motion Equations- Physics Tutorial
What Projectile Motion Actually Is
Projectile motion is just an object moving through the air with only gravity acting on it. No engines, no strings attached. Gravity pulls it down at 9.8 m/s² while it travels forward.
That's it. The whole concept. Everything else in this article is just math describing that simple reality.
The Four Equations You Actually Need
Most textbooks throw five or six equations at you. You only need four. Here's what works:
- Horizontal position: x = v₀ · cos(θ) · t
- Vertical position: y = v₀ · sin(θ) · t - ½gt²
- Horizontal velocity: vₓ = v₀ · cos(θ)
- Vertical velocity: vᵧ = v₀ · sin(θ) - gt
The variables stay constant. The only thing changing is time.
What Each Symbol Means
Don't memorize these wrong. Know what they represent:
- v₀ = initial velocity (how fast you launch it)
- θ (theta) = launch angle (45° gives maximum range on flat ground)
- g = gravitational acceleration (9.8 m/s² on Earth)
- t = time elapsed
- x, y = horizontal and vertical position
The Key Insight Nobody Explains Clearly
Horizontal and vertical motion are completely independent. Gravity only affects the vertical component. The horizontal velocity never changes (ignoring air resistance).
This means:
- Time is the same for both directions
- You calculate horizontal and vertical separately
- You combine them at the end to get total displacement
Maximum Height, Range, and Time
Three values you'll need constantly. Here they are:
- Max height: h = (v₀ · sin(θ))² / (2g)
- Total flight time: T = 2 · v₀ · sin(θ) / g
- Range (flat ground): R = (v₀² · sin(2θ)) / g
Notice range peaks at 45°. That's not a coincidence—it's math.
Comparison: Key Formulas at a Glance
| Quantity | Formula | When to Use |
|---|---|---|
| Horizontal displacement | x = v₀cos(θ)·t | Find where it lands horizontally |
| Vertical displacement | y = v₀sin(θ)·t - ½gt² | Find height at any time |
| Max height | h = v₀²sin²(θ)/2g | Find peak altitude |
| Flight time | T = 2v₀sin(θ)/g | Find total air time |
| Range | R = v₀²sin(2θ)/g | Find horizontal distance |
How to Solve Any Projectile Motion Problem
Stop guessing. Follow these steps every time:
Step 1: Break Down Initial Velocity
Split your launch velocity into components:
- vₓ = v₀ · cos(θ)
- vᵧ = v₀ · sin(θ)
Step 2: Identify What You Know
Write down your known variables. Usually you'll have initial velocity, angle, and either time or displacement.
Step 3: Solve Vertically First
Use vertical equations to find time. Set y = 0 for ground level and solve for t.
Step 4: Plug Time Into Horizontal Equation
Once you have time, find horizontal displacement using x = vₓ · t
Example in 30 Seconds
Ball thrown at 20 m/s at 30°.
vₓ = 20 · cos(30°) = 17.3 m/s
vᵧ = 20 · sin(30°) = 10 m/s
Time to max height: vᵧ/g = 10/9.8 = 1.02s
Total flight time: 2 · 1.02 = 2.04s
Range: 17.3 · 2.04 = 35.3 meters
Common Mistakes That Cost You Points
- Using wrong angle — Make sure you're measuring from horizontal, not from vertical
- Forgetting gravity direction — It's always negative in the vertical equation
- Mixing up velocity and position — vᵧ = v₀sin(θ) - gt gives velocity, not position
- Assuming 45° always gives max range — Only true on flat ground with no air resistance
- Ignoring units — Convert everything to meters and seconds before calculating
Where This Actually Shows Up
You won't calculate cannonball trajectories at work. But the physics shows up in:
- Sports (footballs, basketball free throws, soccer kicks)
- Engineering (designing water fountains, ramps, launching systems)
- Video game physics engines
- Anybody studying for physics exams
The math stays the same. Objects fly in arcs. Gravity pulls them down. You calculate where they land.