(For the purpose of this question, "calculating a collision" means: given the velocities and masses of two objects in a collision, figuring out the new velocities of both objects after the collision).
I know how to calculate a totally elastic collision, and how to calculate a totally inelastic collision.
But I don't know how to calculate a collision which is part elastic and part inelastic. I don't know where to start.
Guidance will be appreciated.
(Go easy on the math please).
There is not such thing as a "partially elastic" collision. Classical collisions between particles can be separated into two categories: elastic and inelastic. Elastic collisions are defined as collisions in which no energy leaves the system (i.e. $E_i = E_f$). All other collisions are inelastic, as some energy is lost ($E_i > E_f$). A perfectly inelastic collision is a type of inelastic collision where all the kinetic energy of the system is lost ($E_f = 0$).
Edit: I should mention that these definitions apply to a CM (center-of-mass) frame of reference. For a non-CM frame, a perfectly inelastic collision becomes one where the maximal amount of kinetic energy is lost. Thanks to David Z. for mentioning this.
Perfectly elastic and perfectly inelastic collisions are just limiting cases on a scale of how much kinetic energy is retained. As noted in @Nathan's answer, if you work in the center-of-mass frame, a perfectly inelastic collision results in 0% of the kinetic energy retained, while perfectly elastic collisions have 100% of kinetic energy retained. So, you just need a way of scaling between those two limiting points.
In all cases, momentum will be conserved. If we choose to work in the CM frame, then it's always zero, and $ v_1m_1 = -v_2m_2 $ before the collision and $ v_1'm_1' = -v_2'm_2' $ after the collision.
Within those constraints, the actual values of $ v_1, v_1', v_2 $ and $ v_2' $ depend on the kinetic energy. KE before and after the collision is
$ KE = (m_1v_1^2 + m_2v_2^2)/2 $
$ KE' = (m_1'v_1'^2 + m_2'v_2'^2)/2 $
Then you choose the degree of inelasticity. If $ \alpha $ is the fraction of kinetic energy that's retained (so $ \alpha = 0 $ corresponds to perfect inelasticity), then
$ KE' = \alpha KE $
Once you've chosen a value for $ \alpha $, you can calculate $ KE' $ from that and the initial velocities. Given known values of $ KE' $ and 0 for kinetic energy and momentum, respectively, you're left with two equations (for net kinetic energy and net momentum) and two unknowns ($ v_1` $ and $ v_2' $), which you can solve algebraically. Then you just need to add and subtract velocities to transform in and out of the CM frame, and you're done.
To parameterize the degree of inelasticity you use the "coefficient of restitution" which is 1 for elastic processes and 0 for completely inelastic processes.
For a two body collision this is described by $$ \text{coef. of restitution} = c_R = \frac{\text{final relative speed}}{\text{initial relative speed}} = \frac{v_2 - v_1}{u_1 - u_2} \,. \tag{*} $$ More generally it is the square root of the fractional bulk kinetic energy.
This also tells you how to compute the final state of such events (assuming the coefficient of restitution is known). You use (*) as an additional constraint on your system just as you would have used equality of kinetic energy for a elastic collision.
From a mathematical standpoint, any collision in which no mass is lost is described by two equations:
Conservation of energy: $ m_1v_1^2 + m_2v_2^2 = m_1v_1'^2 + m_2v_2'^2 + E $
Conservation of momentum: $ m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2' $
You know the masses and initial velocities, so this reduces to a system of two equations in three unknowns ($ v_1', v_2', E $). In order to solve it, you need to know one of those parameters: the energy lost through inelasticity, or one of the final velocities.
