Projectile Motion Calculator
Calculate range, maximum height, time of flight, and velocity components for any launch angle, initial speed, and height. Live trajectory visualization updates as you adjust the angle slider. No signup, runs entirely in your browser.
⏱ 7 min read · Complete guide below
How to Use the Projectile Motion Calculator
- 1Enter the initial speed in m/s and set the launch angle with the slider.
- 2Add an initial height if launching from above ground level (a table, a cliff, a thrower's hand).
- 3Read the range, maximum height, and time of flight, and watch the trajectory redraw live.
- 4Sweep the angle slider to see how range peaks at 45° on flat ground and shifts lower when launching from height.
Worked Example: A Goal Kick at 20 m/s
A soccer ball is kicked from ground level at 20 m/s at a 30° angle. Split the velocity into components: vₓ = 20 cos 30° ≈ 17.3 m/s and v_y = 20 sin 30° = 10 m/s. Gravity only acts vertically, so time of flight is t = 2v_y / g = 20 / 9.81 ≈ 2.04 s. Range is the constant horizontal speed times that time: 17.3 × 2.04 ≈ 35.3 m. Maximum height is v_y² / 2g = 100 / 19.62 ≈ 5.1 m, reached exactly halfway through the flight.
Re-kick the same ball at 45° and the range grows to v² / g = 400 / 9.81 ≈ 40.8 m — the theoretical maximum for that speed. At 60° the range drops back to 35.3 m, identical to the 30° kick, because complementary angles trade hang time for horizontal speed in exactly offsetting amounts. Verify all three angles in the calculator; the pairs of matching ranges fall out of the same equations.
The Big Idea: Two Independent Motions
The key insight that makes projectile motion tractable is that the horizontal and vertical motions are completely independent of each other. Gravity acts only vertically, so it steadily slows, stops, and reverses the up-and-down motion — but it never touches the horizontal motion, which continues at a constant speed the whole time. This is why a bullet fired horizontally and a bullet simply dropped from the same height hit the ground at the same moment: their vertical motions are identical, and the horizontal travel of the fired bullet is irrelevant to how fast it falls. Splitting the launch velocity into its horizontal and vertical components, then treating each separately, is the entire method — and it is exactly what this calculator does.
Why 45° Gives Maximum Range
On flat ground with no air resistance, the greatest horizontal range comes from a 45° launch, and there is a satisfying reason. Range depends on two things that trade against each other: how fast the projectile travels horizontally, and how long it stays in the air. A low angle gives lots of horizontal speed but little hang time; a high angle gives long hang time but little horizontal speed. At 45° these two factors are balanced to give the largest product. The same trade-off explains why complementary angles— like 30° and 60°, or 20° and 70° — produce identical ranges: one sacrifices hang time for speed while the other does the reverse, and the effects cancel out perfectly.
The Idealised Model vs Reality
These equations describe idealised projectile motion, which ignores air resistance and assumes gravity is constant. That model is excellent for heavy, compact, slow-moving objects — a thrown ball, a shot put — and for building physical intuition. But real air resistance (drag) matters a great deal for light objects, high speeds, or spinning balls: it shortens the range, lowers the peak height, and makes the trajectory asymmetric rather than a perfect parabola, with the descent steeper than the ascent. It also nudges the optimal range angle below 45°. Treat the calculator's output as the drag-free baseline — accurate in spirit and often close in practice, but an overestimate of range for anything strongly affected by air.
Projectile Motion Tips
Optimal angle
Set the angle slider to 45° for maximum range on flat ground. Watch how the trajectory becomes flatter at low angles and steeper at high angles, but both give shorter ranges than 45°.
Complementary angles
A launch at 30° and one at 60° land at the same spot. Use both to verify: the calculator should give identical range values for these complementary angles at the same speed.
Horizontal velocity is constant
In ideal projectile motion, vₓ never changes — there is no horizontal force. Only the vertical component changes due to gravity. The trajectory is a parabola.
Real-world applications
Sports (basketball, soccer), artillery ballistics, and satellite orbit insertion all involve projectile motion principles. Air resistance adds complexity but the core equations remain valid for slower objects.
Frequently Asked Questions
What angle gives maximum range?
On flat ground with no air resistance, a 45° launch angle gives the maximum horizontal range. At angles above or below 45°, the range decreases. Complementary angles (e.g. 30° and 60°) give the same range.
What equations govern projectile motion?
Horizontal: x = v₀cos(θ) × t. Vertical: y = h₀ + v₀sin(θ) × t − ½g × t². These assume no air resistance and constant gravity. The calculator uses these to find range, max height, and time of flight.
How does initial height affect range?
Launching from an elevated position (h₀ > 0) increases the total range because the projectile spends more time in the air before hitting the ground. The optimal angle for maximum range shifts below 45° when h₀ > 0.
Does this account for air resistance?
No. This calculator uses idealized projectile motion equations that ignore air drag. In reality, air resistance reduces range and maximum height significantly at high speeds or for low-density objects.
How is time of flight calculated?
The calculator solves the quadratic: −½g × t² + v₀sin(θ) × t + h₀ = 0 for the positive root. This gives the total time from launch until the projectile reaches ground level (y = 0).
Is my data stored?
No. All calculations run locally in your browser. No data is sent to any server.
Why are horizontal and vertical motion treated separately?
Because gravity acts only vertically, the two motions are independent. The horizontal motion continues at a constant speed the whole flight, while the vertical motion is steadily changed by gravity. This independence is why an object dropped and an object thrown horizontally from the same height land at the same time. Splitting the launch velocity into horizontal and vertical components and analysing each on its own is the core method behind every projectile calculation.
Why do 30° and 60° give the same range?
They are complementary angles (they add to 90°), and complementary launch angles always produce the same range on flat ground at a given speed. A lower angle gives more horizontal speed but less hang time, while the higher angle gives more hang time but less horizontal speed — and the two effects exactly offset. The range peaks at 45°, exactly halfway between any such complementary pair, where the trade-off is perfectly balanced.
How does launch height change the best angle?
When you launch from above ground level, the projectile spends extra time falling below the launch point, which rewards horizontal speed a little more than hang time. As a result the optimal angle for maximum range drops below 45° — the higher the launch, the lower the ideal angle. This is why a shot putter or a thrower releasing from shoulder height does best at an angle somewhat under 45° rather than exactly at it.
Does this calculator include air resistance?
No. It uses the idealised, drag-free equations, which assume only gravity acts on the projectile. That is very accurate for heavy, compact, slower objects and ideal for learning the principles, but real air resistance shortens the range and height, makes the descent steeper than the ascent, and lowers the optimal angle. For light objects, high speeds, or spinning balls, treat the results as a drag-free upper bound rather than an exact prediction.
What real-world problems use projectile motion?
Plenty. Sports are full of it — the arc of a basketball shot, a soccer goal kick, a golf drive, or a long jump. It also underlies ballistics and artillery, water fountains and sprinklers, stunt and special-effects planning, and the launch phase of rockets before they leave the atmosphere. The idealised equations here capture the essential behaviour; professional applications then add corrections for drag, wind, and spin.