A runner is jogging in a straight line at a
steady vr= 5 km/hr. When the runner is L=
7.1 km from the finish line, a bird begins flying
straight from the runner to the finish line at
vb= 15 km/hr (3 times as fast as the runner).
When the bird reaches the finish line, it turns
around and flies directly back to the runner.
What cumulative distance does the bird
travel? Even though the bird is a dodo, assume that it occupies only one point in space
(a “zero” length bird), travels in a straight
line, and that it can turn without loss of
speed.
Answer in units of km
The first path, the bird flew 7.1 km
Now on the second path. The bird arrived at his new starting point in time=7.1/15 hr=.47 hr
the runner now is at the 7.1-5km/hr*.47=4.75 mark from the bird. When do they meet?
distance man+distance bird=4.75
speedman*time+speedbird*time=4.75
solve for time final leg....(time)
now find the distance the bird flew on the final run, add the initial 7.1km, and you have it. distance bird final run= time*15km/hr
now add the original 7.1 km, and that is the total bird distance.
the bird reaches the finish line at time t1 = 7.1/15 = 0.4733 hr.
During that time, the runner has moved 2.3667 km to within 4.7333 km of the finish line
From that time on, we need to find t such that they meet again:
4.7333 - 5t = 15t
t = 0.23666
In that time, the bird flew another 3.55 km
In all, the bird flew 10.65 km
To find the cumulative distance traveled by the bird, we need to determine the time it takes for the bird to reach the finish line and then return back to the runner.
First, let's find the time taken for the bird to reach the finish line. Since distance equals speed multiplied by time, we can use the formula:
time = distance / speed
The distance between the runner and the finish line is L = 7.1 km. The bird's speed is vb = 15 km/hr. Plugging in these values, we have:
time = 7.1 km / 15 km/hr
Simplifying this equation, we find:
time = 0.47 hrs
Now let's find the time taken for the bird to return back to the runner. Since the bird is traveling at the same speed as before, the time taken to cover the same distance will be the same:
time = 0.47 hrs
To find the cumulative distance traveled by the bird, we add the distance traveled while going to the finish line and returning back:
cumulative distance = 2 * distance to finish line
cumulative distance = 2 * L
cumulative distance = 2 * 7.1 km
cumulative distance = 14.2 km
Therefore, the cumulative distance traveled by the bird is 14.2 km.