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.
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.
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.