# Trajectory

Establishment of a model for the water bombing from an aircraft

Theoretically, the movement of a spherical drop in an ambient medium depends on :

• the inertial force
• the drag force
• the lift force
• the History force
• the buoyancy force

To simplify our work, we chose to consider only the drag force and the buoyancy force, the other being neglected compared to those two. The balance of the forces gives :

$\left( m_p+C_M*m_G \right) \frac{d \vec {V_p}}{dt}=\left( m_p-m_G \right)+C_D \frac{\pi {R_p}^2}{2} \rho_G \| \vec W - \vec {V_p} \| \left( \vec W - \vec {V_p} \right)$

Then, we deduce the droplet trajectory :

Velocities values

Droplet trajectory in 3D

Droplet trajectory in the $(x,z)$ plan

For each droplet, the trajectory is straight and linear, as shown in the graphics above. The fall time is around 6 seconds, which is realistic compared to the videos of water bombing.

See next : Evaporation