This projectile motion calculator predicts a projectile’s full trajectory from launch speed and angle: maximum height, total flight time, horizontal range, and the velocity components — assuming level ground and no air resistance.
How Projectile Motion Works
A projectile is two independent motions at once: horizontal motion at a constant speed (nothing pushes sideways) and vertical motion under gravity’s constant acceleration — the subject of our Physics collection’s acceleration tools. Splitting the launch velocity into components (vx = v·cosθ, vy = v·sinθ) turns one hard problem into two easy ones, and the paths combine into the famous parabola. This calculator is part of our Physics Calculators collection.
The Trajectory Formulas
- Range: R = v² × sin(2θ) ÷ g
- Max height: H = (v × sinθ)² ÷ (2g)
- Flight time: T = 2 × v × sinθ ÷ g
Worked Example (Verified)
Launch at 20 m/s at 45° on Earth:
- Range: 400 × sin90° ÷ 9.81 = 40.77 m
- Max height: (20 × 0.707)² ÷ 19.62 = 10.19 m
- Flight time: 2 × 14.14 ÷ 9.81 = 2.88 s
- Same throw on the Moon (g = 1.62): range balloons to about 247 m — try it in the widget.
Key Insights (and Where the Model Ends)
- 45° maximizes range on level ground in a vacuum. Angles θ and 90° − θ give equal ranges (30° and 60° land together — one flat and fast, one high and slow).
- Air resistance is ignored here, and in real ballistics it matters hugely — a real baseball loses a large share of the vacuum range. Golf balls, notably, fly farther than the drag-free intuition suggests at moderate speeds thanks to dimples and lift, then still fall short of parabola numbers.
- Launching from a height breaks the symmetry (optimal angle drops below 45°) — this model assumes launch and landing at the same level.
- The everyday-motion sibling for constant speeds is the Speed Calculator.
Related Tools
- Free Fall Calculator — the vertical-only special case
- Kinetic Energy Calculator — launch energy
- Physics Calculators — the full collection
Frequently Asked Questions
What angle gives the maximum range?
45° on level ground without air resistance. With drag or a launch height, slightly less.
What is the formula for projectile range?
R = v² sin(2θ)/g. Range grows with the square of launch speed — doubling speed quadruples range.
Does the projectile’s mass matter?
Not in the vacuum model — gravity accelerates all masses equally. In air, heavier and denser projectiles resist drag better, so mass enters through the back door.
What is the velocity at the top of the arc?
Only the horizontal component remains: vx = v·cosθ. Vertical velocity is momentarily zero — the projectile neither rises nor falls at the apex instant.
How long does a projectile stay in the air?
T = 2·v·sinθ/g for level ground — determined entirely by the vertical component: it is exactly “time to go up and come back down.”
Is this projectile motion calculator free?
Yes — free, instant, no sign-up.