Projectile Motion Calculator
Calculate range, maximum height, time of flight, and trajectory for any projectile. Enter initial velocity, launch angle, and starting height.
Trajectory Summary
Key Values
Flight Time
Trajectory Data Points
| Time (s) | Horizontal (x) | Vertical (y) | Velocity |
|---|
Projectile Motion Formulas
Horizontal: x = v₀·cos(θ)·t
Vertical: y = h₀ + v₀·sin(θ)·t - ½·g·t²
Range: R = v₀²·sin(2θ)/g
Max Height: H = (v₀·sin(θ))²/(2g)
Time of Flight: T = (v₀·sin(θ) + √(v₀²·sin²(θ) + 2g·h₀))/g
Key Concepts
- Horizontal velocity remains constant (no acceleration)
- Vertical velocity changes due to gravity
- Maximum range at 45° (ground level)
- Projectile path is parabolic
- Independent horizontal and vertical motions
Example Calculations
- v₀=50 m/s, θ=45°: Range ≈ 255m
- v₀=30 m/s, θ=60°: Max Height ≈ 34.4m
- v₀=20 m/s, θ=30°: Time of Flight ≈ 2.04s
Frequently Asked Questions
What is projectile motion?
Projectile motion is the motion of an object thrown or projected into the air, subject only to acceleration due to gravity. It follows a parabolic path called a trajectory. The horizontal and vertical motions are independent of each other.
How do you calculate projectile motion?
Step 1: Break initial velocity into horizontal (v₀·cosθ) and vertical (v₀·sinθ) components.
Step 2: Use formulas: x = v₀ₓ·t, y = h₀ + v₀ᵧ·t - ½·g·t²
Step 3: Solve for range, max height, and time of flight.
What is the range of a projectile?
The range is the total horizontal distance traveled. For ground-level launch: R = v₀²·sin(2θ)/g. The maximum range occurs at 45° launch angle.
What is the maximum height of a projectile?
Maximum height is the highest point reached. H = (v₀·sinθ)²/(2g). It occurs when vertical velocity becomes zero.
How do you find time of flight?
Time of flight for ground level: T = 2·v₀·sinθ/g. For elevated launch: T = (v₀·sinθ + √(v₀²·sin²θ + 2g·h₀))/g
Why is 45° the optimal launch angle?
At 45°, sin(2θ) = sin(90°) = 1, which maximizes the range formula R = v₀²·sin(2θ)/g. Any other angle reduces the range.
Does mass affect projectile motion?
In the absence of air resistance, mass does NOT affect projectile motion. All objects fall with the same acceleration due to gravity regardless of mass. Air resistance can affect heavier objects less.
How does initial height affect projectile motion?
Higher initial height increases time of flight and range. The range formula becomes: R = v₀·cosθ·(v₀·sinθ + √(v₀²·sin²θ + 2g·h₀))/g