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
where g ≈ 9.8 m/s² is the acceleration due to gravity and k is the standard basis vector <0,0,1>.

  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.

Latest update 27. November 2008 by Iver Ottosen