Ballistic Trajectory Calculator

Ballistic Trajectory Equation:

\[ y = x \times \tan(\theta) - \frac{g \times x^2}{2 \times v^2 \times \cos^2(\theta)} \]

Horizontal Distance (x):

Launch Angle (θ):

rad

Gravity (g):

m/s²

Initial Velocity (v):

m/s

Vertical Position (y):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Ballistic Trajectory Equation?

The ballistic trajectory equation describes the path of a projectile under the influence of gravity, neglecting air resistance. It calculates the vertical position (y) of a projectile given horizontal distance (x), launch angle (θ), gravity (g), and initial velocity (v).

2. How Does the Calculator Work?

The calculator uses the ballistic trajectory equation:

\[ y = x \times \tan(\theta) - \frac{g \times x^2}{2 \times v^2 \times \cos^2(\theta)} \]

Where:

\( x \) — Horizontal distance (m)
\( \theta \) — Launch angle (radians)
\( g \) — Acceleration due to gravity (m/s²)
\( v \) — Initial velocity (m/s)
\( y \) — Vertical position (m)

Explanation: The equation combines the linear trajectory term (x×tanθ) with the parabolic gravitational term to determine the projectile's height at any horizontal distance.

3. Importance of Ballistic Calculations

Details: These calculations are essential in physics, engineering, military applications, and sports science to predict projectile motion and optimize launch parameters.

4. Using the Calculator

Tips: Enter all values in standard units (meters, radians, m/s). For Earth's gravity, use 9.80665 m/s². Angle must be between 0 and π/2 radians (0-90°).

5. Frequently Asked Questions (FAQ)

Q1: Why is air resistance not included?
A: The basic equation assumes vacuum conditions. For more accurate results with air resistance, numerical methods are required.

Q2: How do I convert degrees to radians?
A: Multiply degrees by π/180 (approximately 0.0174533). Many calculators have a degree-to-radian conversion function.

Q3: What's the maximum height formula?
A: Maximum height occurs at x = (v²×sin(2θ))/(2g) and is given by y_max = (v²×sin²θ)/(2g).

Q4: What's the range equation?
A: Total horizontal range is R = (v²×sin(2θ))/g when launched and landed at the same elevation.

Q5: When is this equation not valid?
A: For very high velocities where air resistance becomes significant, or for very long distances where Earth's curvature matters.