Exercise regarding the Spray Model

A particle flies away from the surface of the fluid at time t = t₁. The surface model gives us the initial velocity vector of the particle v₁ = v(t₁) and its initial position vector r₁ = r(t₁). We want to describe the motion of the particle in 3D-space after it has left the surface. Ignoring air resistance, its acceleration is given by

a(t) = -gk

1. Show that the velocity vector of the particle is given by

   v(t) = -gtk+c

   where c is a constant vector.
2. Show that the position vector of the particle is given by

   r(t) = -½gt²k+ct+d

   where d is a constant vector.
3. Use the initial velocity and position to determine the constants c and d.
4. At time t₂ > t₁, the particle falls back onto the surface or lands outside the fluid body. Describe in words how we can use the vector function r(t) to determine t₂.