Two trains, A and B, start simultaneously from two stations 300 km apart. Train A travels at \(60~km/h\), and Train B travels at \(90~km/h\). If a bird starts flying from Train A towards Train B at \(120~km/h\) and immediately turns back upon reaching Train B, continuing this until the trains meet, what total distance does the bird cover?
Step-by-step Solution:
The key to solving this problem is to realize that the bird's flight duration is exactly the same as the time it takes for the two trains to meet. We do not need to calculate the distance of each of the bird's individual trips.
1. Calculate the time until the trains meet:
The trains are moving towards each other, so their speeds are additive. We can find the time it takes for them to cover the 300 km distance by using their combined relative speed.
Speed of Train A, \(S_A = 60\) km/h.
Speed of Train B, \(S_B = 90\) km/h.
Initial distance = 300 km.
The relative speed of the trains approaching each other is:
\[S_{\text{relative}} = S_A + S_B = 60 + 90 = 150\text{ km/h}\]
Now, we can find the time it takes for them to meet:
\[\text{Time} = \frac{\text{Total Distance}}{\text{Relative Speed}}\]\[\text{Time} = \frac{300 \text{ km}}{150 \text{ km/h}} = 2 \text{ hours}\]
So, the trains will meet each other in exactly 2 hours.
2. Calculate the total distance covered by the bird:
The bird flies continuously for the entire 2 hours that the trains are moving. The speed of the bird is given as 120 km/h.
The total distance covered by the bird is calculated using the formula: Distance = Speed \(\times\) Time.
\[\text{Distance}_{\text{bird}} = \text{Speed}_{\text{bird}} \times \text{Time}\]\[\text{Distance}_{\text{bird}} = 120 \text{ km/h} \times 2 \text{ hours} = 240 \text{ km}\]
Result:
The total distance the bird covers before the trains meet is 240 km.