Projectile Motion Range, Maximum Apex Height & Flight Time Calculator

Compute ballistic trajectory range, apex height, and flight duration.

Horizontal Projectile Range
202.89 m
Maximum Trajectory Height 41.13 m
Total Time of Flight 5.72 s
Horizontal Velocity (v_x) 35.46 m/s
Inputs
m/s
Launch speed (v₀) in meters per second.
deg
Launch elevation angle above horizontal in degrees.
m
Launch height above target ground plane in meters.
m/s²
Gravitational acceleration (9.80665 m/s² Earth standard).
Results & Graph
Horizontal Projectile Range
202.89 m
Maximum Trajectory Height
41.13 m
Total Time of Flight
5.72 s
Horizontal Velocity (v_x)
35.46 m/s

Horizontal Projectile Range

Graph 1 • Initial Velocity Mps Sensitivity Recalculates the domain equation across 50% to 150% of the primary input while holding other parameters constant.
0 m 244.8 m 489.6 m 50%: 52.53 m 52.5 50% 75%: 115.21 m 115.2 75% 100%: 202.89 m 202.9 100% 125%: 315.6 m 315.6 125% 150%: 453.35 m 453.4 150%
Graph 2 • Primary & Supporting Metric Comparison Compares the primary output alongside supporting domain benchmarks.
Horizontal Projectile Range 202.89 m Maximum Trajectory Height 41.13 m Total Time of Flight 5.72 s Horizontal Velocity (v_x) 35.46 m/s
Graph 3 • Exact Output Table & Units Displays each calculated output in its exact domain unit.
Horizontal Projectile Range202.89 mMaximum Trajectory Height41.13 mTotal Time of Flight5.72 sHorizontal Velocity (v_x)35.46 m/s
Summary: Launched at 45 m/s and 38° from 2 m elevation, the projectile reaches an apex of 41.13 m and lands at 202.89 m after 5.72 s.
Summary: Launched at 45 m/s and 38° from 2 m elevation, the projectile reaches an apex of 41.13 m and lands at 202.89 m after 5.72 s.
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Formula & Mathematical Methodology: Projectile Motion Physics & Engineering Equation

vx = v₀·cos(θ)
vy = v₀·sin(θ)
tflight = [vy + vy² + 2·g·h₀] / g
Range = vx × tflight
v_x = v₀·cos(θ); v_y = v₀·sin(θ); t_flight = [v_y + √(v_y² + 2·g·h₀)] / g; Range = v_x × t_flight
  • initialVelocityMps: Initial Velocity Mps (m/s)
  • launchAngleDeg: Launch Angle Deg (deg)
  • initialHeightM: Initial Height M (m)
  • gravityMps2: Gravity Mps2 (m/s²)

Rounding: Physical constants follow CODATA/NIST values; output quantities display to 2 decimal places.

Worked Example: Physics & Engineering Benchmark: Projectile Motion

Compute ballistic trajectory range, apex height, and flight duration.

  1. Enter the physical dimensions, electrical parameters, or thermodynamic state variables.
  2. Apply the governing equation: v_x = v₀·cos(θ); v_y = v₀·sin(θ); t_flight = [v_y + √(v_y² + 2·g·h₀)] / g; Range = v_x × t_flight.
  3. Inspect the primary SI output, derived quantities, and 3 visual SVG graphs.
Result: Computes Projectile Motion via v_x = v₀·cos(θ); v_y = v₀·sin(θ); t_flight = [v_y + √(v_y² + 2·g·h₀)] / g; Range = v_x × t_flight.

Frequently Asked Questions

What equation powers the Projectile Motion Range, Maximum Apex Height & Flight Time Calculator?

It evaluates v_x = v₀·cos(θ); v_y = v₀·sin(θ); t_flight = [v_y + √(v_y² + 2·g·h₀)] / g; Range = v_x × t_flight using IEEE-754 double precision and renders three synchronized SVG plots.

Can I test parameter sensitivity across multiple values?

Yes. Graph 1 automatically plots a 50%–150% parameter response curve, and you can pin up to 3 custom scenarios.

Are all units consistent with SI standards?

Yes. Each input field specifies its exact SI or engineering unit and converts internally where needed.

Science & Engineering · Classical Mechanics & Kinematics

Projectile Motion Range, Maximum Apex Height & Flight Time Calculator: inputs, method, and how to read the result

Compute ballistic trajectory range, apex height, and flight duration.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Initial Velocity Mps: Launch speed (v₀) in meters per second.
  • Launch Angle Deg: Launch elevation angle above horizontal in degrees.
  • Initial Height M: Launch height above target ground plane in meters.
  • Gravity Mps2: Gravitational acceleration (9.80665 m/s² Earth standard).

What the calculation includes

With the default values shown, Horizontal Projectile Range is 202.89 m. The calculation follows Projectile Motion Physics & Engineering Equation; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Initial Velocity Mps from 45 m/s to 50 m/s changes Horizontal Projectile Range from 202.89 m to 249.89 m. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses standard SI physical constants (g = 9.80665 m/s², c = 299,792,458 m/s, σ = 5.670374×10⁻⁸ W·m⁻²·K⁻⁴).
  • Engineering reference model; apply appropriate safety factors and empirical calibration for critical hardware designs.

Scope: SI / NIST Physics & Engineering Standard Check the assumptions and limitations against your situation before relying on the estimate.

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