Under these conditions, from equation 3.5:

if allowance is made for the *added mass*. In this particular case, the equations of motion for the *X *and *Y *directions are mutually independent and therefore can be integrated separately. Furthermore, because the frictional term is now a linear function of velocity, the sign will automatically adjust to take account of whether motion is downwards or upwards. The equations are now integrated, ignoring the effects of *added mass *which can be accounted for by replacing *a *by *a*_ and *b *by *b*_. For the *Y *-direction, integrating equation 3.86 with respect to *t *:

The axes are chosen so that the particle is at the origin at time *t *= 0. If the initial component of the velocity of the particle in the *Y *-direction is *v*, then, when *t *= 0, *y *= 0

and ˙*y *= *v*, and the constant = *v*,

It may be noted that *b/a *= *u*0, the terminal falling velocity of the particle. This equation enables the displacement of the particle in the *Y *-direction to be calculated at any time *t *. For the *X*-direction, equation 3.84 is of the same form as equation 3.86 with *b *= 0. Substituting *b *= 0 and writing *w *as the initial velocity in the *X*-direction, equation 3.88 becomes:

Thus the displacement in the *X*-direction may also be calculated for any time *t *. By eliminating *t *between equations 3.89, 3.90 and 3.91, a relation between the displacements in the *X*– and *Y *-directions is obtained. Equations of this form are useful for calculating the trajectories of particles in size-separation equipment. From equation 3.91: