Projectile Motion Calculator

Projectile Motion is evaluated from Initial Velocity, Launch Angle and Initial Height. The calculation reports Horizontal Range, Maximum Height and Time of Flight.

Results

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About the Projectile Motion Calculator

Projectile Motion is treated here as a quantitative relation between Initial Velocity, Launch Angle and Initial Height and Horizontal Range, Maximum Height, Time of Flight and Horizontal Velocity.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
vₓ = v₀ x cos(θ) - horizontal velocity (constant)
v_y₀ = v₀ x sin(θ) - initial vertical velocity
Max Height = h₀ + v_y₀^2 / (2g)
Time of Flight = [v_y₀ + sqrt(v_y₀^2 + 2g·h₀)] / g
Range = vₓ x Time of Flight
g = 9.80665 m/s^2

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

vₓ = v₀ x cos(θ) - horizontal velocity (constant)
v_y₀ = v₀ x sin(θ) - initial vertical velocity
Max Height = h₀ + v_y₀^2 / (2g)
Time of Flight = [v_y₀ + sqrt(v_y₀^2 + 2g·h₀)] / g
Range = vₓ x Time of Flight
g = 9.80665 m/s^2

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Football Field Goal Kick

Inputs

v0: 28 angle: 45 h0: 0

Horizontal Range: 79.95 m. Maximum Height: 19.99 m. Time of Flight: 4.038 s. Horizontal Velocity: 19.799 m/s. Initial Vertical Velocity: 19.799 m/s

With Initial Velocity = 28, Launch Angle = 45 and Initial Height = 0 as the stated inputs, the result is Horizontal Range = 79.95 m, Maximum Height = 19.99 m and Time of Flight = 4.038 s. Each value corresponds to the declared output fields.

Example 2: Baseball Hit — Home Run Estimate

Inputs

v0: 49.2 angle: 35 h0: 1

Horizontal Range: 233.37 m. Maximum Height: 41.6 m. Time of Flight: 5.79 s. Horizontal Velocity: 40.302 m/s. Initial Vertical Velocity: 28.22 m/s

With Initial Velocity = 49.2, Launch Angle = 35 and Initial Height = 1 as the stated inputs, the result is Horizontal Range = 233.37 m, Maximum Height = 41.6 m and Time of Flight = 5.79 s. Each value corresponds to the declared output fields.

Example 3: Optimal 45° Launch

Inputs

v0: 15 angle: 45 h0: 0

Horizontal Range: 22.94 m. Maximum Height: 5.74 m. Time of Flight: 2.163 s. Horizontal Velocity: 10.607 m/s. Initial Vertical Velocity: 10.607 m/s

With Initial Velocity = 15, Launch Angle = 45 and Initial Height = 0 as the stated inputs, the result is Horizontal Range = 22.94 m, Maximum Height = 5.74 m and Time of Flight = 2.163 s. Each value corresponds to the declared output fields.

Example 4: Cliff Launch — Elevated Start

Inputs

v0: 20 angle: 30 h0: 50

Horizontal Range: 75.72 m. Maximum Height: 55.1 m. Time of Flight: 4.372 s. Horizontal Velocity: 17.321 m/s. Initial Vertical Velocity: 10 m/s

With Initial Velocity = 20, Launch Angle = 30 and Initial Height = 50 as the stated inputs, the result is Horizontal Range = 75.72 m, Maximum Height = 55.1 m and Time of Flight = 4.372 s. Each value corresponds to the declared output fields.

Common Use Cases

Find the range of a ball kicked at an angle
Determine maximum height of a fired projectile
Calculate time of flight for a thrown object