Crossbow Trajectory Calculator

Crossbow Trajectory Equation:

\[ y_{cross\_t} = x_c \cdot \tan(\theta_c) - \frac{g \cdot x_c^2}{2 \cdot v_c^2 \cdot \cos^2(\theta_c)} \]

Distance (x_c):

meters

Launch Angle (θ_c):

radians

Initial Velocity (v_c):

m/s

Gravity (g):

m/s²

Projectile Height (y_cross_t):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Crossbow Trajectory Equation?

The crossbow trajectory equation calculates the vertical position (height) of a projectile at a given horizontal distance, based on launch angle, initial velocity, and gravitational acceleration. This equation is derived from classical projectile motion physics.

2. How Does the Calculator Work?

The calculator uses the crossbow trajectory equation:

\[ y_{cross\_t} = x_c \cdot \tan(\theta_c) - \frac{g \cdot x_c^2}{2 \cdot v_c^2 \cdot \cos^2(\theta_c)} \]

Where:

\( y_{cross\_t} \) — Projectile height at distance x_c (meters)
\( x_c \) — Horizontal distance from launch point (meters)
\( \theta_c \) — Launch angle (radians)
\( v_c \) — Initial velocity (m/s)
\( g \) — Gravitational acceleration (9.81 m/s²)

Explanation: The equation accounts for both the vertical component of initial velocity and the effect of gravity on the projectile's trajectory.

3. Importance of Trajectory Calculation

Details: Accurate trajectory calculation is crucial for archery, ballistics, and projectile sports. It helps predict where a projectile will land and optimize launch parameters for target hitting.

4. Using the Calculator

Tips: Enter distance in meters, launch angle in radians, initial velocity in m/s, and gravitational acceleration (default 9.81 m/s²). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: Why use radians instead of degrees for angle?
A: Radians are the standard unit for trigonometric functions in physics equations. 1 radian ≈ 57.3 degrees.

Q2: What is a typical crossbow velocity?
A: Modern crossbows typically have velocities between 50-150 m/s, depending on draw weight and bolt weight.

Q3: How does air resistance affect the calculation?
A: This equation assumes no air resistance. In reality, air resistance reduces range and alters trajectory, especially at higher velocities.

Q4: What's the maximum range calculation?
A: Maximum range occurs at 45° launch angle. The range equation is: \( R = \frac{v_c^2 \cdot \sin(2\theta_c)}{g} \)

Q5: Can this be used for other projectiles?
A: Yes, this equation applies to any projectile under constant gravity with no air resistance, including arrows, bullets, and thrown objects.