Projectile Range Calculator: equations, variables, units and worked solution

The projectile range calculator calculates Horizontal range, Time of flight, Maximum height, Horizontal velocity from Launch speed, Launch angle, Launch height, Gravity. It does not hide the arithmetic: the result panel shows the governing formula, canonical-unit conversion, numeric substitution, unrounded evaluation and final rounded answer for every output.

Variables and measurement units

For the Projectile Range Calculator, Launch speed, Launch angle, and Launch height are normalized to m/s, °, and m before speed*cos(angle*pi/180)*((speed*sin(angle*pi/180)+sqrt((speed*sin(angle*pi/180))**2+2*gravity*launchHeight))/gravity) is evaluated. The unrounded value used for Horizontal range is retained internally; formatting is applied only to the result cards.

Formula model

Horizontal and vertical velocity components determine flight time, range, maximum height and impact conditions.

Formula-based worked derivation

1. Horizontal range
   1. Formula: Horizontal range = speed × cos(angle × pi ÷ 180) × ((speed × sin(angle × pi ÷ 180)+sqrt((speed × sin(angle × pi ÷ 180))^2+2 × gravity × launchHeight)) ÷ gravity)
   2. Default substitution: Horizontal range = (25) × cos((45) × pi ÷ 180) × (((25) × sin((45) × pi ÷ 180)+sqrt(((25) × sin((45) × pi ÷ 180))^2+2 × (9.80665) × (1.5))) ÷ (9.80665))
   3. Unrounded evaluation: 65.1985294339 m
   4. Displayed answer: 65.1985 m
2. Time of flight
   1. Formula: Time of flight = (speed × sin(angle × pi ÷ 180)+sqrt((speed × sin(angle × pi ÷ 180))^2+2 × gravity × launchHeight)) ÷ gravity
   2. Default substitution: Time of flight = ((25) × sin((45) × pi ÷ 180)+sqrt(((25) × sin((45) × pi ÷ 180))^2+2 × (9.80665) × (1.5))) ÷ (9.80665)
   3. Unrounded evaluation: 3.68818578289 s
   4. Displayed answer: 3.6882 s
3. Maximum height
   1. Formula: Maximum height = launchHeight+(speed × sin(angle × pi ÷ 180))^2 ÷ (2 × gravity)
   2. Default substitution: Maximum height = (1.5)+((25) × sin((45) × pi ÷ 180))^2 ÷ (2 × (9.80665))
   3. Unrounded evaluation: 17.4330658278 m
   4. Displayed answer: 17.4331 m
4. Horizontal velocity
   1. Formula: Horizontal velocity = speed × cos(angle × pi ÷ 180)
   2. Default substitution: Horizontal velocity = (25) × cos((45) × pi ÷ 180)
   3. Unrounded evaluation: 17.6776695297 m/s
   4. Displayed answer: 17.6777 m/s

In the Projectile Range Calculator, changing Launch speed, Launch angle, and Launch height rebuilds the numeric substitution for speed*cos(angle*pi/180)*((speed*sin(angle*pi/180)+sqrt((speed*sin(angle*pi/180))**2+2*gravity*launchHeight))/gravity). The engine converts selected measurements to m/s, °, and m, retains unrounded values, and, when reverse solving is available, inserts the solved variable back into the same equation to verify Horizontal range with a numerical residual.

Input and output interpretation

Use measured or documented values for Launch speed, Launch angle, Launch height, Gravity. The calculated outputs are Horizontal range, Time of flight, Maximum height, Horizontal velocity. Check each intermediate line before relying on the final value; an implausible intermediate quantity usually identifies a unit, range or assumption error.

Algorithm and verification

The Projectile Range Calculator uses its own `projectile-range` JavaScript engine to calculate Horizontal range from Launch speed, Launch angle, and Launch height with speed*cos(angle*pi/180)*((speed*sin(angle*pi/180)+sqrt((speed*sin(angle*pi/180))**2+2*gravity*launchHeight))/gravity). Calculator-owned boundary and mode tests check that workflow; an external frozen-reference oracle checks the numeric outputs, and physical source mutation testing confirms that a changed `projectile-range` engine is rejected.

Visual interpretation

In the Projectile Range Calculator, this guidance applies to Launch speed, Launch angle, and Launch height and the reported Horizontal range. The `projectile-range` workflow evaluates speed*cos(angle*pi/180)*((speed*sin(angle*pi/180)+sqrt((speed*sin(angle*pi/180))**2+2*gravity*launchHeight))/gravity) in m/s, °, and m; The trajectory model neglects aerodynamic drag, wind, spin and terrain. Use a specialized exterior-ballistics model where those effects matter.. Verification context: 354efa.

Dimensional formula audit

The projectile range calculator normalizes speed (Launch speed, m/s), angle (Launch angle, °), launchHeight (Launch height, m), gravity (Gravity, m/s²) before evaluating the equations. Its reported quantities are Horizontal range in m, Time of flight in s, Maximum height in m, Horizontal velocity in m/s. This separation matters because a numerical value without its measurement dimension can produce a plausible-looking but physically or financially incorrect answer. Conversion factors are applied before substitution, and output conversion occurs only after the canonical result has been calculated at full precision.

• Horizontal range = speed × cos(angle × pi ÷ 180) × ((speed × sin(angle × pi ÷ 180)+sqrt((speed × sin(angle × pi ÷ 180))^2+2 × gravity × launchHeight)) ÷ gravity)
• Time of flight = (speed × sin(angle × pi ÷ 180)+sqrt((speed × sin(angle × pi ÷ 180))^2+2 × gravity × launchHeight)) ÷ gravity
• Maximum height = launchHeight+(speed × sin(angle × pi ÷ 180))^2 ÷ (2 × gravity)
• Horizontal velocity = speed × cos(angle × pi ÷ 180)

Formula sensitivity and boundary verification

To verify the projectile range calculator, hold all other inputs fixed and change speed within its permitted range. The live substitution line shows exactly where that value enters the equation for Horizontal range. Repeat the check with angle. The result must follow the displayed algebra, remain finite, and retain the stated output unit. At minimum and maximum boundaries, the validator rejects undefined domains, impossible denominators and nonphysical values rather than silently returning a number.

Manual reproduction of the result

For an independent hand check, first convert every selected unit to the canonical units shown in the variable table. Next copy the governing equation, replace each symbol with the canonical value shown in the live derivation, and calculate the intermediate expression without early rounding. Finally round only once to the displayed precision and compare both the numerical value and unit with the calculator card. This process makes the projectile range calculator reproducible instead of relying on an unexplained result.

Limitations

The trajectory model neglects aerodynamic drag, wind, spin and terrain. Use a specialized exterior-ballistics model where those effects matter.