Projectile Range Calculator

Related Calculators

Horizontal range (distance): How far it travels horizontally until it reaches the landing level (vertical position y = 0 measured from the landing surface) [1].

Time of flight: Total time from launch until it reaches the landing level [1].

Launch speed v0: The starting speed (magnitude) right at launch, before splitting into horizontal and vertical parts [1].

Launch angle: The direction of the launch measured above the horizontal; the calculator may convert degrees to radians for trig [1].

Launch height y0: The launch point height compared to the landing surface: y0 = 0 is flat ground, positive is above, negative is below.

Gravity g: Downward acceleration used in the ideal model (treated as constant during the flight) [2].

Initial horizontal speed vx0: The horizontal part of v0; in ideal projectile motion it stays constant because there is no horizontal acceleration [1].

Initial vertical speed vy0: The vertical part of v0; gravity changes it over time [1].

Impact speed (ideal, no drag): The speed right before reaching the landing level, computed without air resistance (so it depends on the height difference y0).

Mistake: Entering radians when the angle field is being treated as degrees (or the other way around).
Fix: Check Angle input unit in Advanced options, then type the angle in that same unit.

Mistake: Using Solve for launch speed v0 or Solve for launch angle while Launch height above landing level y0 (selected unit) is not 0.
Fix: Set y0 to 0 for solve modes, or switch back to Range (distance) mode for nonzero height problems.

Mistake: Mixing unit systems (for example choosing US/Imperial but typing speed in m/s or height in meters).
Fix: Match your inputs to the selected Unit system , or use Speed input unit (Advanced options) if your speed is given in mph or km/h.

Mistake: Typing 9.81 for gravity while using feet and ft/s (US/Imperial).
Fix: In US/Imperial, gravity near Earth is about 32.174 ft/s^2, or choose a Gravity preset and keep the same Unit system .

Mistake: Expecting a landing result when the projectile cannot reach the landing level (often happens with very negative y0 and a small upward speed).
Fix: Increase Launch speed v0 , increase the upward part of the launch (angle), make y0 less negative, and read the Notes card for the exact math reason.

Mistake: Thinking Impact speed at landing (ideal, no drag) must always equal the launch speed.
Fix: If y0 is 0 they match; if y0 is positive (launch above landing), impact speed is higher, and if y0 is negative it can be lower (or N/A if landing is unreachable).

No air resistance: The calculator assumes ideal projectile motion (no drag, no wind, no spin effects). Real objects usually land closer than the ideal range, especially if they are light or fast.

Constant gravity: It treats Gravity g (selected unit per second squared) as constant and straight downward, which is a good approximation for typical classroom-sized problems.

Landing level is fixed at y = 0: All results are computed for the moment the projectile reaches the landing surface level (y = 0). Your Launch height above landing level y0 (selected unit) is measured from that same level.

Must be physically reachable: For the general (nonzero y0) case, the math needs a real square root term. If not, there is no real time when the projectile reaches the landing level with the given inputs.

Solve-for modes are flat-ground only: Solving for speed or angle uses the simplified flat-ground relationship, so it requires y0 = 0. If y0 is not 0, you can still use Range mode and adjust inputs.

Angle and direction limits: If you enter an angle outside the allowed range shown by the input rules, the calculator will stop and ask you to fix it. Very small angles (near 0) or near-vertical angles (near 90 degrees) can give near-zero range.

This calculator uses ideal projectile motion: constant downward gravity, no air resistance, and constant horizontal velocity [1].

θ (angle in radians) = θ (angle in degrees) * (π/180)

vx0 (initial horizontal speed) = v0 * cos(θ)

vy0 (initial vertical speed) = v0 * sin(θ)

To find when it reaches the landing level (y = 0), solve the vertical motion equation for time. The calculator uses the positive-time root:

t (time of flight) = (vy0 + sqrt(vy02 + 2*g*y0)) / g

This requires g > 0 and the inside of the square root v_y0² + 2*g*y₀ to be at least 0 (otherwise the projectile never reaches the landing level).

R (horizontal range) = vx0 * t

h (max height above launch) = vy02 / (2*g)

H (max height above landing) = y0 + h

For impact speed at the landing level (still ideal, no drag), it uses an energy/kinematics result [1]:

v (impact speed) = sqrt(v02 + 2*g*y0)

If v₀² + 2*g*y₀ is negative, impact speed is N/A because the landing level is unreachable.

Solve-for modes (flat ground only, y0 = 0) use the common same-height range relationship [2]:

R = (v02 * sin(2*θ)) / g

v0 = sqrt((R*g) / sin(2*θ))

s = (R*g) / v02; 2*θ = arcsin(s) or (π - arcsin(s))

Mini example (flat ground): let v0 = 20 m/s, angle = 45 degrees, y0 = 0 m, g = 9.81 m/s^2. Then v_x0 = 14.142 m/s and v_y0 = 14.142 m/s. Time of flight t = (2*v_y0)/g = 2.883 s, range R = v_x0*t = 40.77 m, and max height above launch h = v_y0²/(2*g) = 10.19 m.