calculate_projectile_with_drag
Calculate realistic projectile trajectories with air resistance, spin effects, wind, and variable air density for sports balls and other objects.
Instructions
Calculate projectile motion including air resistance (drag).
Uses numerical integration (RK4) to solve motion equations with:
- Quadratic drag force: F_drag = 0.5 * ρ * v² * Cd * A
- Magnus force (spin effects): F_magnus = 0.5 * ρ * Cl * A * ω * r * v
- Wind effects (constant wind vector)
- Variable air density (altitude and temperature effects)
This provides REALISTIC trajectories for sports balls, projectiles,
and other objects moving through air or water. Compare with
calculate_projectile_motion (no drag) to see dramatic differences!
Common drag coefficients (Cd):
- Sphere: 0.47 (default)
- Baseball: 0.4
- Golf ball: 0.25 (dimples reduce drag)
- Football (American): 0.05-0.15 (orientation-dependent)
- Basketball: 0.55
- Soccer ball: 0.25
- Skydiver (belly-down): 1.0-1.3
- Streamlined car: 0.25-0.35
Args:
initial_velocity: Launch velocity in m/s
angle_degrees: Launch angle in degrees (0-90)
mass: Object mass in kg
cross_sectional_area: Cross-section perpendicular to motion in m²
initial_height: Launch height in meters (default 0)
drag_coefficient: Drag coefficient Cd (default 0.47 for sphere)
fluid_density: Fluid density in kg/m³ (air=1.225, water=1000)
gravity: Gravitational acceleration m/s² (default 9.81)
time_step: Integration time step in seconds (default 0.01)
max_time: Maximum simulation time in seconds (default 30)
spin_rate: Spin rate in rad/s for Magnus force (default 0, no spin)
spin_axis: Spin axis unit vector [x, y, z] (default [0, 0, 1] = vertical)
wind_velocity: Wind velocity [vx, vy] in m/s (default [0, 0], no wind)
altitude: Altitude above sea level in meters (default 0, affects air density)
temperature: Air temperature in Celsius (default 15, affects air density)
Returns:
Dict containing:
- max_height: Maximum altitude reached (m)
- range: Horizontal distance traveled (m)
- time_of_flight: Total flight time (s)
- impact_velocity: Speed at landing (m/s)
- impact_angle: Angle at landing (degrees below horizontal)
- trajectory_points: [[x, y], ...] for plotting
- energy_lost_to_drag: Energy dissipated by drag (J)
- initial_kinetic_energy: Initial KE (J)
- final_kinetic_energy: Final KE (J)
- lateral_deflection: Lateral deflection from spin/wind (m)
- magnus_force_max: Maximum Magnus force magnitude (N)
- wind_drift: Total wind drift (m)
- effective_air_density: Effective air density used (kg/m³)
Example - Baseball curveball (2500 rpm backspin):
result = await calculate_projectile_with_drag(
initial_velocity=40.23, # 90 mph
angle_degrees=10,
mass=0.145,
cross_sectional_area=0.0043,
drag_coefficient=0.4,
spin_rate=261.8, # 2500 rpm = 261.8 rad/s
spin_axis=[0, 0, 1] # Backspin (vertical axis)
)
# Backspin increases range and height!
Example - Golf ball at altitude (Denver, 1600m):
result = await calculate_projectile_with_drag(
initial_velocity=70,
angle_degrees=12,
mass=0.0459,
cross_sectional_area=0.00143,
drag_coefficient=0.25,
altitude=1600, # Denver elevation
temperature=20 # Summer day
)
# Less air resistance = longer drive!
Example - Soccer free kick with wind:
result = await calculate_projectile_with_drag(
initial_velocity=25,
angle_degrees=15,
mass=0.43,
cross_sectional_area=0.0388,
drag_coefficient=0.25,
wind_velocity=[5, 0], # 5 m/s tailwind
spin_rate=50, # Sidespin for curve
spin_axis=[0, 1, 0] # Horizontal axis
)
# Wind drift + Magnus curve!Input Schema
| Name | Required | Description | Default |
|---|---|---|---|
| initial_velocity | Yes | ||
| angle_degrees | Yes | ||
| mass | Yes | ||
| cross_sectional_area | Yes | ||
| initial_height | No | ||
| drag_coefficient | No | ||
| fluid_density | No | ||
| gravity | No | ||
| time_step | No | ||
| max_time | No | ||
| spin_rate | No | ||
| spin_axis | No | [0, 0, 1] | |
| wind_velocity | No | [0, 0] | |
| altitude | No | ||
| temperature | No |