At t=0 a ball is thrown straight upward from the ground level at speed vo. at the same instant, a distance d above, a ball is thrown straight downward, also at speed vo.

where do the balls collide in terms of d, vo, and g?

1) Upward motion

y1 = Vo*t - g(t^2) / 2

2) Downward motion

y2 = D - [Vo*t  + g(t^2) / 2]

3) Collision => y1 = y2

      y1 = Vo*t - g(t^2) / 2
      y2 = D - [Vo*t + g(t^2) / 2]

Vo*t - g(t^2) / 2 = D - [Vo*t + g(t^2) / 2]

Vo*t - g(t^2) / 2 = D - Vo*t - g(t^2)/2

D = 2Vo*t => t = D / (2Vo)

Substitute the value of t in the equation of y1 (it is the same if you do it in the equation of y2)

y1 = Vo*t - g(t^2) / 2 = Vo [D/(2Vo) ] - g [D / (2Vo)]^2  / 2

y1 = D/2 - g(D^2) / 8(Vo ^2)

  y1 = D/2 - g(D^2) / 8(Vo ^2)

Alinear relationship would look like a straight line on the graph. it could have a positive negative or 0 slope. so it could be a line potentially going in any direction but it must be perfectly straight.