Is a trajectory calculator the same as a projectile motion calculator?
Yes — they are two names for the same calculation. A trajectory is the curved path a projectile follows, and projectile motion is the physics that produces it. Either way you are solving the same set of equations: horizontal velocity stays constant while vertical velocity changes under gravity, and combining the two gives a parabola. This page handles horizontal, angled and purely vertical launches, returning range, maximum height, time of flight and impact speed, with the trajectory drawn as you change the inputs.
What are the three types of projectile motion?
In precise physics terms there are three standard types. A horizontal projectile is thrown flat (launch angle 0°) from a height, so its initial vertical velocity is zero. An oblique (angular) projectile is launched at an angle to the horizontal, tracing a parabolic arc — the most general case. A vertical projectile is fired straight up (launch angle 90°) with no horizontal motion, so its range is zero. This calculator handles all three, with an animated ball that flies the real computed path for each.
What is projectile motion?
Projectile motion is the curved path an object follows when thrown or launched near the Earth, moving under gravity alone (ignoring air resistance). The horizontal velocity stays constant while gravity constantly pulls the object down. For an oblique (angular) launch this produces a parabolic arc; a horizontal projectile traces the falling half of that parabola; and a vertical projectile moves in a straight line up and back down.
What are the projectile motion equations?
The core projectile motion formulas (ignoring air resistance) are: horizontal velocity vₓ = v₀·cos θ; initial vertical velocity v_y = v₀·sin θ; time of flight T = (v_y + √(v_y² + 2gh)) / g; range R = vₓ·T; maximum height H = h + v_y²/(2g); and impact speed v = √(vₓ² + v_yf²) with v_yf = v_y − g·T. For a horizontal launch θ = 0° (so v_y = 0 and T = √(2h/g)); for a vertical launch θ = 90° (so vₓ = 0 and range = 0). Here v₀ is the launch speed, θ the launch angle, h the launch height, and g = 9.81 m/s² on Earth.
How do you calculate the range of a projectile?
Range is horizontal velocity times time of flight: Range = vₓ × T, where vₓ = v₀·cos θ. For an oblique launch from the ground this simplifies to Range = v₀²·sin(2θ)/g, with maximum range at a 45° launch angle. A vertical projectile has zero range (no horizontal velocity), while a horizontal projectile launched from a height travels farther the higher it starts, since the flight lasts longer.
What launch angle gives the maximum range?
For an oblique (angular) projectile launched from ground level, 45° gives the maximum range because sin(2θ) peaks at 2θ = 90°. Angles equally above and below 45° — say 30° and 60° — give the same range. When launching from a height, the optimal angle drops below 45° (the extra fall time rewards a flatter, faster shot). A vertical projectile (90°) maximizes height rather than range.
How is the time of flight and impact speed found?
Time of flight comes from solving the vertical equation h + v_y·t − ½g·t² = 0 for the positive root: T = (v_y + √(v_y² + 2gh)) / g. Impact speed combines the horizontal and final vertical velocities: v = √(vₓ² + v_yf²), where v_yf = v_y − g·T. By energy conservation, an object launched and landing at the same height returns at exactly its launch speed; launched from a height, it lands faster.