Projectile Motion Calculator
Try a launch at 20 m/s and 45 degrees from level ground in the projectile motion calculator. Add a launch height to see the range grow and the best angle drop below 45.
Range
- Range Horizontal distance to where it lands back at ground level: horizontal speed × time in the air = 14.14 × 2.884 = 40.79 m. Horizontal speed never changes, because gravity only acts vertically.
- 40.79 m
- Maximum height Launch height plus vy² / 2g = 0 + 14.14² / (2 × 9.80665). At the top the vertical velocity is zero; it is v² = u² + 2as with v = 0.
- 10.2 m
- Time of flight On level ground this is 2·vy / g: the climb and the fall take the same time.
- 2.884 s
- Time to the highest point vy / g: the time for gravity to cancel the vertical velocity.
- 1.442 s
- Launch velocity components u·cos θ and u·sin θ at 45 degrees. Splitting the launch into these two is the entire method: the horizontal part is constant velocity, the vertical part is free fall.
- horizontal 14.14 m/s, vertical 14.14 m/s
- Impact speed √(u² + 2gh). It does not depend on the angle; that is energy conservation in one line. From ground level it lands at its launch speed.
- 20 m/s
- Impact angle
- 45° below horizontal
- Angle for the longest range Exactly 45 degrees from level ground, and any two angles that add to 90 give the same range.
- 45°, reaching 40.79 m
| Launch angle | Time in the air | Range | Max height |
|---|---|---|---|
| 15° | 1.056 s | 20.39 m | 1.366 m |
| 30° | 2.039 s | 35.32 m | 5.099 m |
| 45° (longest) | 2.884 s | 40.79 m | 10.2 m |
| 60° | 3.532 s | 35.32 m | 15.3 m |
| 75° | 3.94 s | 20.39 m | 19.03 m |
- Air resistance is ignored, as it is in every textbook projectile problem. Real projectiles fall short of these ranges, a light or fast one by a lot, and the best angle drops below 45 degrees.
- Gravity is standard gravity, 9.80665 m/s², and is taken as constant and straight down, which is right for anything that stays near the ground. Times are in seconds.
How the Kinematics Calculator works
Projectile motion is two constant-acceleration problems running at the same time. Split the launch velocity u at angle θ into a horizontal part, u·cos θ, and a vertical part, u·sin θ. Horizontally nothing pushes, so the horizontal speed never changes. Vertically the object is in free fall with acceleration g. The time is shared between them, and that is the whole method.
The formulas, from level ground: time of flight T = 2u·sin θ / g, maximum height H = (u·sin θ)² / 2g, and range R = u²·sin 2θ / g.
Worked example, the one above: 20 m/s at 45 degrees. Both components are 20 × 0.7071 = 14.14 m/s. Time of flight is 2 × 14.14 / 9.80665 = 2.884 s, maximum height is 14.14² / (2 × 9.80665) = 10.20 m, and the range is 20² × sin 90° / 9.80665 = 40.79 m.
Why 45 degrees throws furthest. The range depends on sin 2θ, which peaks at 1 when 2θ = 90°. Any two angles that add to 90 (30 and 60, 15 and 75) give the same range, as the table shows: the steep one goes high and spends longer in the air, the shallow one travels faster sideways for less time, and the products come out equal.
Launching from a height changes that. The object now has extra time to fall, the time-of-flight equation becomes the positive root of h + u·sin θ·t − ½gt² = 0, and the best angle drops below 45 degrees, because a flatter launch spends that extra time moving sideways. The calculator finds the optimum exactly: tan θ = u / √(u² + 2gh). A shot put released at head height is the classic case.
The impact speed is √(u² + 2gh) and does not depend on the angle at all. That is energy conservation: whatever the path, the object has fallen the same net height.
What this does not model, and it matters more here than anywhere else on the page: air resistance. Drag acts against the velocity, cutting the range and pulling the best angle well below 45 degrees for fast or light projectiles. A thrown ball, an arrow or a golf drive goes markedly shorter than these figures, and spin adds lift that this model has no term for. Treat the answer as the vacuum limit: correct for the textbook problem, an upper bound in a field. Gravity is standard gravity, constant and straight down, which is right for anything that stays near the ground and travels a few kilometers or less.
Kinematics Calculator FAQ
What is the formula for the range of a projectile?
On level ground with no air resistance, R = u²·sin 2θ / g, where u is the launch speed and θ the launch angle. At 20 m/s and 45 degrees that is 400 / 9.80665 = 40.79 m. From a height, the range is horizontal speed times the full time of flight.
What angle gives the maximum range?
45 degrees from level ground, ignoring air. From a height the optimum is lower, given by tan θ = u / √(u² + 2gh). With real air resistance it is lower again, which is why long throws and kicks are launched well under 45 degrees.
How do you find the maximum height of a projectile?
Use only the vertical velocity, u·sin θ. At the top it is zero, so v² = u² + 2as gives H = (u·sin θ)² / 2g above the launch point. The horizontal speed plays no part in how high it goes.