A runner is jogging in a straight line at a
steady vr= 7.5 km/hr. When the runner is
L= 6.3 km from the finish line, a bird begins
flying straight from the runner to the finish
line at vb= 22.5 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.
016 (part 2 of 2) 10.0 points
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.
part 2 is easier than part 1
The runner takes 6.3/7.5 = 0.84 hr to reach the finish line.
The bird flies all during that time at a speed of 22.5 km/hr
distance = speed * time, right?
For part 1, the bird takes 6.3/22.5 = 0.28 hr to reach the finish line
During that time, the runner has moved 7.5 * 0.28 km closer, so he is only 6.3-2.1 = 4.2 km from the finish line.
So now find when the runner's distance and the bird's distance adds up to 4.2 km
4.2 - 7.5t = 22.5t
Then add that distance to the bird's starting 6.3 km
To find the cumulative distance that the bird travels, we need to consider both the distance it travels to the first encounter and the distance it travels on each round trip from the runner to the finish line.
First, let's calculate the distance the bird travels to the first encounter with the runner. The initial distance between the runner and the finish line is L = 6.3 km. Since the bird is 3 times faster than the runner, its speed is vb = 22.5 km/hr.
Using the formula: distance = speed * time, we can determine the time it takes for the bird to cover the initial distance to the first encounter.
time = distance / speed
time = 6.3 km / 22.5 km/hr
time = 0.28 hr
Now, let's calculate the distance the bird travels on each round trip from the runner to the finish line. The total distance the bird covers on each round trip is twice the distance between the runner and the finish line (2 * L).
The bird repeats this round trip until the runner reaches the finish line. So, the number of round trips is given by the ratio of the total distance remaining for the runner to the distance covered on each round trip (L / (2 * L)).
Cumulative distance traveled by the bird = distance traveled to the first encounter + (number of round trips) * (distance covered on each round trip)
Cumulative distance traveled = 6.3 km + (L / (2 * L)) * (2 * L)
Simplifying the expression, we get:
Cumulative distance traveled = 6.3 km + (1/2) * (2 * L)
Cumulative distance traveled = 6.3 km + L
Substituting the value of L:
Cumulative distance traveled = 6.3 km + 6.3 km
Cumulative distance traveled = 12.6 km
Therefore, the cumulative distance traveled by the bird, including the distance traveled to the first encounter, is 12.6 km.
To find out how far the bird travels from the beginning, we need to consider the distance it travels in each round trip between the runner and the finish line.
In the first round trip, the bird flies from the runner to the finish line, which is a distance of L = 6.3 km. It then turns around and flies back to the runner, covering the same distance of L = 6.3 km. So, in the first round trip, the bird travels a total distance of 6.3 + 6.3 = 12.6 km.
In each subsequent round trip, the bird again travels the same distance back and forth. Therefore, the distance traveled by the bird in each round trip remains constant at 12.6 km.
Now, let's calculate the number of round trips the bird makes. The runner's speed is vr = 7.5 km/hr, and the bird's speed is vb = 22.5 km/hr.
The time taken by the runner to cover the initial distance L = 6.3 km is:
Time = Distance / Speed = 6.3 km / 7.5 km/hr = 0.84 hours.
During this time, the bird travels at its speed of vb = 22.5 km/hr. So, the distance the bird covers during this time is:
Distance = Speed * Time = 22.5 km/hr * 0.84 hr = 18.9 km.
Since each round trip distance is 12.6 km, the bird makes 18.9 km / 12.6 km = 1.5 round trips.
Therefore, the total distance traveled by the bird from the beginning, including the distance traveled to the first encounter, is:
Total distance = (Number of round trips + 1) * Round trip distance
Total distance = (1.5 + 1) * 12.6 km = 3.5 * 12.6 km = 44.1 km.
So, the bird travels a cumulative distance of 44.1 km from the beginning.