Example 3: A troll is hiding 15 m under a bridge when Billy Goat Gruff trots over. Now trolls don't like goats much and this is a particularly nasty troll! The troll throws a stone upward at 20 m/s hoping to hit Billy. Billy catches the movement out of the corner of his eye and, a half second later, spits out a tin can he's been chewing on (at 10 m/s downward). At what point do the two projectiles pass each other?

Solution:

I'll take the origin at the troll and positive direction up. It helps to summarize the information...

troll	goat
v_o = 20 m/s s_o = 0 t_T = t a = - 9.8 m/s²	v_o = -10 m/s s_o = 15 m t_G = t - 0.5

Subtracting the second (Goat) equation from the first, and collecting terms, the squared terms in t subtract out and I get

Substituting this back into the troll equation,