Question

suppose that the position functions of 2 bugs B1 and B2 in motion along the same line are B1=1/2t^2-t+3 and B2=-1/4t^2+t+1 for t is greater than or equal to zero.
a) prove the bugs will not collide
b)how close do the bugs get to each other?

I would greatly appreciate if you could help me I am very confused and unsure how to solve this problems.

Answer 1

if the bugs collide, then at some point B1 = B2. That means the distance between the bugs is zero, so

B1-B2 = 0
(1/2 t^2-t+3)-(-1/4t^2+t+1)=0
3/4 t^2 - 2t + 2 = 0

Since the discriminant is negative, B1-B2 has no real roots. That is, it is never zero. So, the bugs do not collide.

B1-B2 has a minimum at t = 4/3
So plug it in and see how close they get.