Two trains, each having a speed of'34 km,/h, are headed at each other on the same straight track. A bird that can fly 58 km /h flies ofi" the front of one train when they are 102 kmapart and heads directly for the other train. On reaching theother train it flies directly back to the first train, and so forth. (a) How many trips can the bird make from one train to the other before they crash‘? (b) What is the total distance thebird travels?
To solve this problem, let's break it down step by step:
(a) To determine the number of trips the bird can make before the trains crash, we need to calculate the time it takes for the two trains to collide.
Firstly, we should find the relative speed of the two trains. Since they are heading towards each other, we can add their speeds:
Relative speed = Speed of first train + Speed of second train
Relative speed = 34 km/h + 34 km/h = 68 km/h
Now we can calculate the time it takes for the trains to collide by using the formula:
Time = Distance / Speed
The distance between the two trains is given as 102 km.
Time = 102 km / 68 km/h ≈ 1.5 hours
Since the bird is flying at a constant speed, we can use the time it takes for the trains to collide to determine how many trips the bird can make.
During each round trip, the bird covers a distance equal to the distance traveled by one train:
Distance covered in one round trip = Speed of bird × Time
Distance covered in one round trip = 58 km/h × 1.5 hours = 87 km
However, we need to remember that the bird only starts flying when the trains are 102 km apart. Therefore, the number of trips the bird can make is the distance between the trains divided by the distance covered in one round trip:
Number of trips = Distance between the trains / Distance covered in one round trip
Number of trips = 102 km / 87 km ≈ 1.17 trips
Since we can't have a fraction of a trip, we can conclude that the bird can make 1 trip from one train to the other before they crash.
(b) To find the total distance the bird travels, we need to multiply the distance covered in one round trip by the number of trips made:
Total distance = Distance covered in one round trip × Number of trips
Total distance = 87 km × 1 trip = 87 km
Therefore, the total distance traveled by the bird is 87 km.