Projectile Motion Calculator

Frequently Asked Questions

What is the maximum range of a projectile?

On level ground with no air resistance, range is maximized at 45°. Range = v₀² × sin(2θ) / g. For a 45° launch at 20 m/s: range ≈ 40.8 m.

Does mass affect projectile motion?

In a vacuum, no: gravity accelerates all masses equally at 9.81 m/s². With air resistance, lighter or larger objects slow more from drag.

How do I split velocity into components?

vₓ = v₀ cos θ (horizontal, constant) and vᵧ = v₀ sin θ (vertical, changes with gravity). Treat them independently and recombine at landing.

What is the time of flight?

On level ground, t = 2 v₀ sin θ / g. Maximum height = (v₀ sin θ)² / (2g). Range = horizontal velocity × time of flight.

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How This Calculator Works

Projectile motion is the textbook example of separating a problem into independent horizontal and vertical components: horizontal velocity stays constant while gravity acts only on the vertical axis. Enter an initial velocity, a launch angle between 0° and 90°, and an optional launch height, and this tool resolves the velocity into its horizontal and vertical parts, then applies the standard kinematic equations to compute range, maximum height, total time of flight, time to the peak, impact speed, and the two velocity components. Gravity is fixed at g = 9.81 m/s² (Earth's surface) and air resistance is ignored, so results describe the idealized vacuum trajectory.

Formula / Method

The launch angle θ splits the speed: horizontal v_x = v₀ cosθ, vertical v_y = v₀ sinθ. Time of flight from a launch height h₀ solves h₀ + v_yt − ½gt² = 0, giving T = [v_y + √(v_y² + 2gh₀)] ÷ g. Range R = v_x × T. Time to peak t_peak = v_y ÷ g, and maximum height H = h₀ + v_y² ÷ (2g). Impact speed combines the components at landing: √(v_x² + (v_y − gT)²). For level ground the range simplifies to R = v₀² sin(2θ) ÷ g.

Worked Example

Launch at 30 m/s, 45°, from ground level (h₀ = 0). Components: v_x = 30 cos45° ≈ 21.21 m/s, v_y = 30 sin45° ≈ 21.21 m/s. Time of flight T = 2v_y ÷ g = 2 × 21.21 ÷ 9.81 ≈ 4.32 s. Range R = v_x × T ≈ 21.21 × 4.32 ≈ 91.7 m, matching v₀² sin(90°) ÷ g = 900 ÷ 9.81. Max height H = v_y² ÷ (2g) = 450 ÷ 19.62 ≈ 22.9 m, reached at t_peak ≈ 2.16 s. With no launch height, impact speed equals the launch speed, 30 m/s.

Common Mistakes

Forgetting to resolve the velocity: you cannot use the full speed in the vertical equation; only v₀ sinθ fights gravity.

Using the level-ground range formula when launching from a height: a nonzero h₀ extends flight time and range.

Working in degrees inside sine and cosine without converting to radians, when the math library requires radians.

Assuming 45° is always optimal: that only holds for equal launch and landing heights with no drag.

Expecting real-world accuracy: this is a vacuum model; air resistance shortens range substantially for light or fast projectiles.

Tips & FAQ

Why does 45° maximize range? On flat ground without drag, range depends on sin(2θ), which peaks when 2θ = 90°, i.e. θ = 45°: the angle that best balances horizontal speed against time aloft. With air drag the optimum drops to roughly 30 to 40 degrees. Does launch height change the best angle? Yes: launching from above the landing surface lowers the optimal angle below 45°. Real-world uses? Basketball (a ~55° arc gives a larger entry window through the rim than a flat ~35° shot), soccer free kicks (the baseline arc follows these equations, with spin adding Magnus curve), and ballistics (artillery adds corrections for drag, wind, and the Coriolis effect). Does gravity vary? Slightly: g falls about 0.03% per 1,000 m of altitude, negligible for everyday problems, so this tool fixes it at 9.81 m/s². Why is impact speed sometimes higher than launch speed? When you launch from a height, the projectile gains vertical speed falling the extra distance, so it lands faster than it left. See also the kinetic energy calculator to convert that impact speed into joules.

Projectile Motion Calculator

Frequently Asked Questions

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