Calculate range, maximum height, time of flight, and impact velocity for projectile motion from initial velocity, launch angle, and optional initial height.
Projectile motion describes the curved path of an object launched into the air, subject only to gravity. The motion splits into independent horizontal (constant velocity) and vertical (constant acceleration) components. This calculator determines range, maximum height, flight time, and trajectory for any launch angle and speed.
Equations of projectile motion
A projectile launched at speed v₀ and angle θ has initial horizontal velocity v₀cos(θ) and initial vertical velocity v₀sin(θ). Horizontal distance is x = v₀cos(θ)t. Vertical position is y = v₀sin(θ)t − ½gt². The time of flight (returning to launch height) is T = 2v₀sin(θ)/g.
Maximum height occurs at the midpoint of flight: H = v₀²sin²(θ)/(2g). The horizontal range is R = v₀²sin(2θ)/g. Range is maximised at θ = 45° when launch and landing heights are equal. For unequal heights, the optimal angle shifts below 45°.
R = v₀² × sin(2θ) / g
Range R is the horizontal distance for a projectile returning to its launch height.
H = v₀² × sin²(θ) / (2g)
Maximum height reached by the projectile above the launch point.
T = 2 × v₀ × sin(θ) / g
Total flight time for a projectile returning to launch height.
Real-world factors
The ideal projectile equations assume no air resistance and flat terrain. In practice, drag reduces range significantly — a baseball hit at 45° in a vacuum would travel about 170 m, but in air it travels only about 120 m. Wind, spin (Magnus effect), and altitude (lower air density) further modify the trajectory.
Worked example and interpretation
A worked example helps translate the projectile motion calculator maths into a realistic scenario so the user can compare the headline result with a concrete set of inputs.
That matters because a result is easier to trust when the page shows how the same logic behaves in a practical case instead of leaving the formula abstract.
Frequently asked questions
Why is 45 degrees the optimal launch angle for maximum range?
Range depends on sin(2θ), which reaches its maximum value of 1 when 2θ = 90°, i.e. θ = 45°. At this angle, the initial velocity is equally split between horizontal and vertical components, balancing flight time and horizontal speed. With air resistance, the optimal angle drops to roughly 38–42° because drag penalises higher trajectories more.
How does launch height affect the range?
Launching from an elevated position increases flight time because the projectile has farther to fall. This extra time adds horizontal distance, increasing range. The optimal launch angle also decreases below 45° when the launch point is above the landing point.
How can I check the projectile motion calculator result manually?
The safest manual check is to follow the same formula or rule one step at a time and compare that working with the calculator output. That catches sign errors, bracket mistakes, and input-order mixups without requiring any extra method beyond the underlying maths itself.
