Question

How do you determine what the final height of two objects in the air will be after a perfectly inelastic collision?

Answer 1

first, you need to calculate the combined mass velocity after the collision to do that apply momentum conservation with the direction included

suppose two masses are Ma and Mb having velocity Va and Vb initially and after  collision suppose they together have speed = Vc so from momentum conservation

Pi = Pf

Ma*Va + Mb*Vb = ( Ma+Mb) *Vc

so Vc = (Ma*Va + Mb*Vb) / ( Ma+Mb)

now apply kinematics equtions

v^2 = u^2 +2*ah

at maximum height v = 0

so  h_max = u^2/2*g = (Vc)^2/ (2*g)

at this way u can determine what the final height of two objects in the air

let me know in a comment if there is any problem or doubts