(A)A runner is jogging at a steady 7.7 km/hr. when the runner is 3.9 km. form the finish line, a bird begins flying from the runner to the finish line at 46.2 km/hr (6 times as fast as the runner). When the bird reaches the finish line, it turns around and flies back to the runner. How far does the bird travel? Even though the bird is a dod, assume that it occupies only one point in space and that it can turn without loss of speed.
(B) 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).
Flight time of bird to finish line, FL, = 3.9/46.2 = .0844 hr.
Distance runner travels during this time period = .0844(7.7) = .65km.
Distance from runner to FL = 3.9 - .65 = 3.25km.
Net closing speed between runner and bird = 7.7 + 46.2 = 53.9Km/hr.
Time for runner and bird to meet = 3.25/53.9 = .060 hr.
Distance covered by runner = .060(7.7) = .462km.
Distance covered by bird to runner = .060(46.2) = 2.785km.
Total distance traveled by bird up to first runner-bird meeting = 3.9 + 2.785 = 6.685km.
Remaining distance of runner to FL = 3.9 - .65 - .462 = 2.788km.
Time of runner to FL = 2.788/7.7 = .362 hr.
During this time period, the bird flies back and forth between the runner and the FL at the constant speed of 46.2km/hr covering a total distance of 46.2(.362) = 16.728km.
Therefore, the total distance traveled by the bird from the start = 6.685 + 16.728 = 23.413km.
I hope I didn't slip a digit or two.
Your problem brings to mind a golden oldie from the field of recreational mathematics that you might find amusing.
Two trains 150 miles apart are traveling toward each other along the same track. The first train goes 60 miles per hour; the second train rushes along at 90 miles per hour. A fly is hovering just above the nose of the first train. It buzzes from the first train to the second train, turns around immediately, flies back to the first train, and turns around again. It goes on flying back and forth between the two trains until they collide. If the fly's speed is 120 miles per hour, how far will it travel?
We want to know the total distance that the fly covers, so let's use Distance = Rate * Time to solve the problem. We already know the fly's rate of flight. If we can find the time that the fly spends in the air, we can figure out how far it travels.
Ignore the fly for a minute, and concentrate on the trains. The first train is traveling at 60 miles/ hour and the second train is going 90 miles/ hour, so they are approaching each other at 60 miles/ hour + 90 miles/ hour = 150 miles/ hour. Now we know the rate at which the trains are closing in on each other and their distance apart (150 miles), so we can find the time until they crash:
Distance = Rate * Time
Time = Distance / Rate
= (150 miles) / (150 miles/ hour)
= 1 hour.The fly spends the same amount of time traveling as the trains. It goes 120 miles/ hour, so in the one hour the trains take to collide, the fly will go 120 miles.
To solve these problems, we need to consider the relative speeds and distances traveled by the runner and the bird.
(A) The bird is flying 6 times as fast as the runner, which means the ratio of their speeds is 46.2 km/hr : 7.7 km/hr = 6:1. We can use this ratio to determine the time it takes for the bird to reach the finish line.
Let's denote the distance traveled by the bird as "d". Since the runner is 3.9 km from the finish line when the bird starts, the total distance the bird needs to cover is (3.9 km + d).
Since time = distance / speed, the time taken by the bird to reach the finish line is (3.9 km + d) / 46.2 km/hr.
At this point, the bird turns around and flies back to the runner at the same speed. So the total distance traveled by the bird in this scenario is (3.9 km + d) + (3.9 km + d).
(B) In this scenario, the bird continues to repeat the back and forth trips until the runner reaches the finish line. The bird's distance traveled includes not only the distance to the first encounter but also the additional distance covered in subsequent trips.
Let's consider the distance traveled by the bird in each round trip. In the first round trip, the bird flies a distance of 2*(3.9 km + d). In the second round trip, the bird covers a distance of 2*(3.9 km + d) again. This pattern continues until the runner reaches the finish line.
To determine the total distance traveled by the bird, we need to sum up the distances in each round trip until the runner reaches the finish line.
Now that we have explained the approach, we can use these formulas to calculate the distances traveled by the bird in both scenarios.