Work out the flight of a projectile launched at an angle. Enter the launch speed, the angle, and (optionally) the starting height, and the calculator returns the time of flight, the range, the maximum height, and the velocity components.
The projectile motion calculator works out the complete trajectory of an object thrown, kicked, or launched into the air at an angle. Enter the launch speed, the launch angle, and the starting height, and it returns the time of flight, the range (how far it travels horizontally), the maximum height it reaches, and the horizontal and vertical components of the launch velocity. It is the go-to tool for physics homework, for sports and ballistics questions, and for anyone curious about how far and how high a launched object will go.
Projectile motion is the curved path an object follows when it is launched near the Earth's surface and moves under the influence of gravity alone (ignoring air resistance). It is one of the classic topics in mechanics because it neatly combines two simple motions — constant-velocity motion sideways and free-fall motion up and down — into a single, elegant parabola. This calculator applies the standard equations of projectile motion so you can predict the whole flight from just a few inputs.
Projectile motion is the motion of an object launched into the air and then acted on only by gravity. A thrown ball, a kicked football, a fired cannonball, and a jet of water from a hose all follow projectile paths. The defining feature is that the motion splits cleanly into two independent parts: a horizontal motion at constant velocity (because nothing pushes or slows the object sideways) and a vertical motion that is exactly free fall (because gravity pulls it down). Combining these two produces the familiar curved, parabolic trajectory.
The key insight — and the reason projectile problems are solvable with simple formulas — is that the horizontal and vertical motions do not affect each other. Gravity changes only the vertical velocity; the horizontal velocity stays constant throughout the flight. The calculator handles both components and combines them to give the results you care about.
Everything starts by splitting the launch velocity V₀ into components using the launch angle α:
horizontal velocity: Vₓ = V₀ × cos(α)
vertical velocity: Vₔ = V₀ × sin(α)
For a launch from ground level (initial height h = 0), the results simplify to:
time of flight: t = 2 × V₀ × sin(α) ÷ g
maximum height: h max = V₀² × sin²(α) ÷ (2g)
range: R = V₀² × sin(2α) ÷ g
When the projectile is launched from a starting height h above the landing level, the flight lasts longer and the general formulas apply:
time of flight: t = [V₀sin(α) + √(V₀²sin²(α) + 2gh)] ÷ g
maximum height: h max = h + V₀²sin²(α) ÷ (2g)
range: R = V₀cos(α) × t
The calculator automatically uses the general formulas, which reduce to the simple ground-level ones when you set h to zero.
Suppose a ball is launched from the ground at 20 m/s at an angle of 45°, with g = 9.80665 m/s². The velocity components are both V₀cos(45°) = V₀sin(45°) ≈ 14.14 m/s. Then:
So the ball is in the air for about 2.88 seconds, rises about 10.2 meters, and lands about 40.8 meters away. Raising the launch height would extend both the flight time and the range.
You can sanity-check any of these figures against the individual motions: the vertical velocity of 14.14 m/s divided by gravity gives about 1.44 seconds to reach the peak, and doubling that gives the 2.88-second total flight, since the rise and fall take equal time for a ground-level launch. The horizontal velocity, meanwhile, never changes throughout the flight — it is the steady 14.14 m/s that carries the ball its 40.8-meter range. Breaking a result apart like this is the best way to build intuition for how the pieces fit together.
A famous result of projectile motion is that, for a launch from ground level, the range is greatest at a launch angle of 45°. That is because range depends on sin(2α), which reaches its maximum value of 1 when 2α = 90°, i.e. α = 45°. Angles equally spaced above and below 45° give the same range — for instance, 30° and 60° produce identical horizontal distances (though the higher angle gives a longer, taller flight). This symmetry is easy to explore with the calculator: try several angles at the same speed and watch the range peak at 45°. Note, though, that the 45° rule assumes launch and landing at the same height and no air resistance; when the projectile starts from an elevated position, the optimal angle for maximum range drops slightly below 45°.
Understanding why projectile motion works comes down to treating the two directions separately. Horizontally, there is no force (ignoring air resistance), so the object moves at a constant speed V₀cos(α) for the entire flight; the horizontal distance is simply this speed multiplied by the time. Vertically, the object behaves exactly like something thrown straight up: it starts with an upward speed V₀sin(α), slows under gravity, momentarily stops at the top (which is the maximum height), then falls back down, speeding up again. The time to reach the top is V₀sin(α)/g, and for a ground-level launch the descent takes the same time, giving the total flight time of 2V₀sin(α)/g. Splitting the problem this way turns a curved, two-dimensional path into two simple one-dimensional motions.
This calculator models ideal projectile motion, in which gravity is the only force and the path is a perfect parabola. In reality, air resistance (drag) acts against the motion, shortening the range, lowering the maximum height, and making the descending part of the path steeper than the ascending part — so the trajectory is no longer symmetric. For dense, slow, or short-range projectiles — a shot put, a thrown stone, a short kick — drag is small and the idealized results are very accurate. For light, fast, or long-range objects — a golf ball, an arrow, a bullet — air resistance matters a great deal, and the real range will be noticeably less than the calculator predicts. Treat the results as an upper bound and a solid first approximation.
The launch-height feature matters whenever a projectile does not start and finish at the same level — a ball thrown from a rooftop, a cannon on a hill, or a shot put released from shoulder height. Suppose a stone is thrown at 10 m/s at 30° from a cliff 5 m high. The vertical launch speed is Vₔ = 10 × sin(30°) = 5 m/s. Because the stone lands 5 m below where it started, the flight lasts longer than a ground launch would: t = [5 + √(5² + 2 × 9.80665 × 5)] ÷ 9.80665 ≈ 1.64 s. The horizontal speed is Vₓ = 10 × cos(30°) ≈ 8.66 m/s, so the range is 8.66 × 1.64 ≈ 14.2 m, and the maximum height (measured from the ground below) is 5 + 5²/(2 × 9.80665) ≈ 6.27 m. Notice that the extra 5 m of launch height both stretches the flight time and pushes the landing point farther out — which is exactly why throwing from an elevated position increases range. Enter these numbers into the calculator to see each result confirmed instantly.
Few places show projectile motion more vividly than sport. A basketball free throw is a projectile problem: the shooter must choose a launch speed and angle so the ball's parabola passes through the hoop, and because the ball is released above the ground and lands at rim height, the ideal angle is a little under the textbook 45°. A shot put or javelin is thrown from shoulder height, so athletes and coaches use projectile formulas (plus corrections for air resistance) to find the release angle that maximizes distance — typically in the low-to-mid 30s of degrees rather than 45°, precisely because of the launch height. In golf, the launch angle and ball speed off the clubface determine carry distance, and in soccer or American football, the arc of a long kick is a projectile trajectory whose hang time (the time of flight this calculator computes) is as tactically important as its distance. While real sports balls are affected by spin and drag, the ideal projectile model captures the essential trade-offs — higher angle for more height and hang time, lower angle for a flatter, faster path — and it is the starting point every coach and physics student reasons from. This calculator lets you explore those trade-offs by changing the speed, angle, and release height and watching how the range, height, and flight time respond.
For a ground-level launch, range = V₀² × sin(2α) ÷ g. For a launch from a height, range = V₀cos(α) × time of flight. This calculator uses the general formula automatically.
From ground level, 45° gives the greatest range, because range depends on sin(2α), which peaks at α = 45°. Launching from a height lowers the optimal angle slightly below 45°.
For a ground launch, time of flight = 2 × V₀ × sin(α) ÷ g. With an initial height h, use t = [V₀sin(α) + √(V₀²sin²(α) + 2gh)] ÷ g.
Maximum height = h + V₀² × sin²(α) ÷ (2g), where h is the launch height. From the ground (h = 0) it is simply V₀²sin²(α)/(2g).
Yes — for a ground-level launch, complementary angles (such as 30° and 60°) produce the same range, because sin(2α) is the same for both. The higher angle produces a taller, longer-lasting flight.
No. It models ideal projectile motion with gravity as the only force, giving a perfect parabola. Real projectiles experience drag and fall somewhat short of these figures, especially light or fast objects.
This Projectile Motion Calculator is provided for educational and general informational purposes. It models idealized projectile motion (no air resistance) using the standard kinematic equations. Real-world results differ due to drag and other factors; do not use it for safety-critical decisions.