Math Help - Basic Kinematics

A runner is jogging at a steady vr =
3:8 km=hr. When the runner is L = 5:1 km
from the finish line, a bird begins flying
from the runner to the finish line at vb =
11:4 km=hr (3 times as fast as the runner).
When the bird reaches the finish line, it turns
around and flies back to the runner. Even
though the bird is a dodo, we will assume
that it occupies only one point in space, i.e., a
zero length bird.

How far does the bird travel? Answer in
units of km.


I'm way lost here. So far, I figure when the bird reaches the finish line, the bird will have traveled 5.1km while the human will have traveled 1.7km, leaving him 3.4km from the finish. I don't know where to go from here though, it seems like guess and check is the only way but I know that's not it

*edit* Ehh should've put this in high school math, my bad :\

Hey

Vr=3.8km/h
Vb=11.4km/h
L=5.1km

Velocity=distance/time
time b reach finish:
vb=d/t
11.4=5.1/t
t = 0.44737hr*60mins
=26.8421mins

d=vr*t R distance in that time
=3.8*0.44737
=1.7km

distance left
5.1km - 1.7km =3.4km

Find the time that the runner and bird meet:
You know the total distance and both there distances should add to that total
3.4 = distB + distR (substitute d=v*t)
3.4 = (vb*t)+(vr*t)
3.4 = (11.4*t)+(3.8*t) solve for t
3.4 = 15.2t
t = 0.2237hrs *60min
= 13.4210mins

Total B dist:
dtotal = (.44737*11.4)+(.2237*11.4)
= 7.65km

Therefore the bird traveled a total of 7.65km

Wow man thanks, it makes a ton of sense now. I never figured that the distance they would meet at would equal 3.4km, but it makes perfect sense when you think about it

It's a two part equation, here's the second part:

After this first encounter, the bird then turns
around and flies from the runner back to the
finish line, turns around again and flies back
to the runner. The bird repeats the back and
forth trips until the runner reaches the finish
line.
How far does the bird travel from the be-
ginning? (i.e., include the distance traveled
to the first encounter) Answer in units of km.

I'll just continue solving and hopefully everything works out nicely, I'll post with my answer in a sec. Thanks again

I'm getting about 17.80km total for the bird's distance, sound about right? It almost seems like this could be a limit problem or something since the differences are almost negligible at the end.

*EDIT* Damn, I typed in 17.8km for my answer and got it wrong. All that plug and chug for nothing

There must be some sort of way to streamline this equation down though, I performed about 20 of the same calculations over and over again and I don't think that's what the teacher had in mind.

Wow, there was a MUCH easier way to solve part two.

All I had to do was find out how long it took for the human to reach the finish, then I multiplied that time by the bird's velocity. DONE!

Special thanks goes out to winet from #math on EFnet for helping me get this one