Projectile Motion Simulator
Launch a ball at any speed, angle and height — on Earth, the Moon, Mars or Jupiter — and watch its path, velocity components and the exact numbers behind it.
A projectile keeps a constant horizontal velocity while gravity pulls it down, so its path is a parabola. From level ground, range R = u² sin 2θ / g, maximum height H = u² sin²θ / 2g and time of flight T = 2u sin θ / g. Launching at 20 m/s and 45° on Earth gives R = 400 ÷ 9.8 = 40.8 m.
Launch settings
Drag the sliders or type exact values.
Trajectory
Both axes use the same scale, so the path has its true shape.
Air resistance is ignored (ideal projectile). Real balls fly a little shorter and lower.
Compare launches
Every launch is kept in its own colour (up to 8).
| Path | u (m/s) | θ | h (m) | g | Time | Max height | Range |
|---|
Formulas with your numbers
Two motions, one parabola
A projectile moves sideways at a steady speed while gravity pulls it down at a steady rate. Put the two together and you get a parabola.
Set speed and angle
Use the sliders or type exact values — the results and the dashed preview update instantly.
Pick the gravity
Compare Earth with the Moon, Mars and Jupiter, or enter your own g.
Launch and compare
Each launch stays on the screen in a new colour so you can compare ranges and heights.
Check the formulas
Every result is worked out step by step with your numbers.
Split the velocity
uₓ = u cos θ uy = u sin θ
The horizontal part never changes (no air resistance). The vertical part falls by g every second — that is why the trail dots are evenly spaced sideways.
Launched from the ground (h = 0)
T = 2u sin θ / g
H = u² sin²θ / 2g
R = u² sin 2θ / g — largest at θ = 45°.
Launched from a height h
T = [u sin θ + √(u² sin²θ + 2gh)] / g
Then R = u cos θ × T and H = h + u² sin²θ / 2g. From a height, the best angle is a little less than 45°.
Complementary angles
From level ground, θ and (90° − θ) give the same range because sin 2θ = sin (180° − 2θ). For example 30° and 60°. The steeper shot goes higher and stays in the air longer.
Projectile Motion Simulator — your questions
At what angle is the range of a projectile maximum?
From level ground the range is greatest at 45°, because R = u² sin 2θ / g and sin 2θ reaches its maximum value of 1 when 2θ = 90°. If the projectile is launched from a height above the ground, the best angle becomes a little less than 45°, since the extra fall gives it more time to travel forward.
What are the formulas for projectile motion?
Split the launch velocity into uₓ = u cos θ and u_y = u sin θ. From level ground, time of flight T = 2u sin θ / g, maximum height H = u² sin²θ / 2g and range R = u² sin 2θ / g. The horizontal velocity never changes; only the vertical part is altered by gravity, falling by g every second.
Why do 30° and 60° give the same range?
Because the range depends on sin 2θ, and sin 60° equals sin 120°. Any two angles that add up to 90° — called complementary angles — give the same range from level ground. The steeper one, 60°, reaches a greater maximum height and stays in the air longer, while the flatter one gets there faster.
Why is the path of a projectile a parabola?
The horizontal distance grows steadily with time, x = uₓt, while the vertical drop grows with the square of time, y = u_y t − ½gt². Eliminating t between the two gives an equation in which y depends on x², and that is the equation of a parabola. Two uniform motions combined always produce this curve.
What is the velocity of a projectile at the highest point?
At the highest point the vertical component of velocity is zero, but the horizontal component is unchanged. So the speed there is u cos θ, directed horizontally — never zero, unless the object was thrown straight up. The acceleration at that instant is still g downwards, which is why the projectile immediately starts falling.
What happens to a projectile on the Moon?
The Moon's gravity is 1.62 m/s², about six times weaker than Earth's 9.8 m/s². Since range, height and time of flight are all inversely proportional to g, the same launch gives roughly six times the range and height and stays airborne about six times as long. There is no air on the Moon, so nothing slows it down.
Does this simulator include air resistance?
No — it models the ideal projectile taught in Class 11, where air resistance is ignored and the only force acting is gravity. A real ball meets a drag force that grows with speed, so it falls short of the calculated range, reaches a slightly lower maximum height and follows a path that is steeper coming down than going up.