Trace the exact arc of a launched object.
Set a launch speed, an angle, and a starting height, then watch the full flight path form. The calculator splits the motion into a steady horizontal part and a gravity-driven vertical part, then reports how far the object travels, how high it climbs, how long it stays in the air, and how fast it is moving when it lands.
The launch
What this flight means
The path the object follows
Height against horizontal distance
Flight scorecard
Range at different launch angles
Same speed and height, changing the angle. On flat ground the range peaks at 45 degrees, and a pair of angles that add up to 90 degrees give the same distance.
| Angle | Range (m) | Max height (m) | Flight time (s) | Range vs now |
|---|
Projectile motion, explained
Horizontal and Vertical Motion Are Independent
The key idea behind projectile motion is that the sideways and up-down parts of the flight do not affect each other. The horizontal velocity vx = v0 x cos(theta) stays constant the entire time because nothing pushes the object sideways. The vertical velocity vy = v0 x sin(theta) is slowed, stopped, and reversed by gravity. Both share the same clock, so the time the object spends rising and falling is exactly the time it has to travel forward. That single shared time links how high it goes to how far it lands.
Range, Height, and Time of Flight
Time of flight comes from the vertical motion: T = (vy + sqrt(vy squared + 2 x g x h0)) / g, which is when the object returns to the landing surface. Multiply that by the constant horizontal speed to get range: R = vx x T. Peak height is H = h0 + vy squared / (2g), reached at the halfway point in time for a level launch. From a ground-level launch these reduce to the familiar forms R = v0 squared x sin(2 theta) / g and H = v0 squared x sin squared(theta) / (2g).
Why 45 Degrees Gives the Farthest Throw
On flat ground the range depends on sin(2 theta), which is largest when 2 theta equals 90 degrees, so theta equals 45 degrees. A neat result follows: any two angles that add to 90 degrees, such as 30 and 60, produce the same range. One path is high and slow, the other flat and fast, yet they land at the same spot. When the launch starts above the landing height, the ideal angle drops slightly below 45 degrees because the extra fall gives low, fast throws more time in the air.
What the Model Leaves Out
This calculator uses the classic no-drag model, so the path is a clean parabola and the landing speed from ground level equals the launch speed. Real projectiles meet air resistance, which shortens range, lowers the peak, and pulls the best angle well below 45 degrees for light or fast objects like a thrown ball. Wind, spin (the Magnus effect on a curveball), and changes in gravity with altitude also shift the outcome. For dense, slow objects over short distances the simple model is very close to reality.
Common questions
What angle gives the maximum range?
From ground level on flat terrain, 45 degrees gives the farthest range because range depends on sin(2 x angle), which peaks at 45 degrees. If the launch starts above the landing height, the best angle drops a little below 45 degrees.
Why do 30 and 60 degrees land at the same distance?
For a ground-level launch, any two angles that add up to 90 degrees produce the same range. One arc is high and slow, the other is flat and fast, but both cover the same horizontal distance.
Does the object land at the same speed it was launched?
Yes, if it launches and lands at the same height, energy conservation means the landing speed equals the launch speed. If it starts higher than it lands, it hits the ground faster because the extra drop adds vertical speed.
Does mass affect the trajectory?
No, in this no-drag model the path depends only on launch speed, angle, height, and gravity. A heavy and a light object launched identically follow the exact same arc. Mass only matters once air resistance is included.
How does air resistance change the result?
Air resistance shortens the range, lowers the peak height, and reduces the landing speed. It also pushes the best launch angle below 45 degrees, often to 30 to 40 degrees for a thrown ball. This calculator ignores drag, so treat its numbers as the ideal vacuum case.
What gravity value should I use?
Use 9.81 m/s squared for Earth at sea level. For other bodies enter 1.62 for the Moon or 3.72 for Mars. Higher gravity means shorter, lower, quicker flights, lower gravity means the object hangs and travels much farther.
Uses the classical constant-gravity model with no air resistance, so the arc is a perfect parabola. Real throws lose range and height to drag, wind, and spin, more so for light or fast objects. The ground is treated as flat. Estimates for learning and planning only.