A runner is jogging in a straight line at a
steady vr= 6.8 km/hr. When the runner is
L= 2.2 km from the finish line, a bird begins
flying straight from the runner to the finish
line at vb= 20.4 km/hr (3 times as fast as
the runner). When the bird reaches the finishline, 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.
Part 2
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 beginning (including the distance traveled to the
first encounter)?
Answer in units of km
To find the cumulative distance the bird travels, we need to consider its motion in both directions - from the runner to the finish line and back.
In the first scenario, the bird is flying towards the finish line while the runner is still running. The bird's speed is three times faster than the runner, so its speed is 3 * vr = 3 * 6.8 km/hr = 20.4 km/hr.
To determine the time it takes for the bird to reach the finish line, we need to find the time it takes for the runner to cover the remaining distance L = 2.2 km. We can use the equation:
time = distance / speed
Substituting the values, we have:
time = L / vr = 2.2 km / 6.8 km/hr ≈ 0.3235 hr
During this time, the bird continues flying towards the finish line at its speed of 20.4 km/hr. Therefore, the distance the bird travels in this scenario is:
distance = speed * time = 20.4 km/hr * 0.3235 hr ≈ 6.604 km
Now, for the second scenario, the bird turns around and flies back to the runner. Since the bird is already at the finish line, it will fly back the same distance it just traveled. Therefore, the distance the bird travels in this scenario is also approximately 6.604 km.
To find the cumulative distance the bird travels in both scenarios, we add the distances:
cumulative distance = distance in scenario 1 + distance in scenario 2 = 6.604 km + 6.604 km = 13.208 km
So, the cumulative distance the bird travels is approximately 13.208 km.
For Part 2, we know that after the first encounter, the bird keeps repeating the back and forth trips until the runner reaches the finish line.
In each trip, the bird covers a distance of 2 * 6.604 km = 13.208 km (the distance to the finish line and back). Since the runner is L = 2.2 km away from the finish line, the bird needs to make ceil(L / 13.208) trips to cover the remaining distance, where ceil(x) represents the smallest integer greater than or equal to x.
ceil(L / 13.208) = ceil(2.2 km / 13.208 km/trip) = ceil(0.1669) = 1
Therefore, the bird needs to make one additional trip to cover the remaining distance.
Hence, the total distance the bird travels from the beginning (including the distance traveled to the first encounter) is:
total distance = (number of trips + 1) * distance per trip
= (1 + 1) * 13.208 km
= 26.416 km
So, the bird travels a total distance of approximately 26.416 km.
time = distance/speed
So, how long does the runner take to reach the finish line.
Now use that time to get the bird's distance: distance = speed * time