Answer:
1)
75 kmh⁻¹
2)
75 kmh⁻¹
Explanation:
1)
= Speed of train from station A to station B = 80 kmh⁻¹
= distance traveled from station A to station B
= time of travel between station A to station B
we know that
![Time = \frac{distance}{speed}](https://tex.z-dn.net/?f=Time%20%3D%20%5Cfrac%7Bdistance%7D%7Bspeed%7D)
![t_{ab} = \frac{d_{ab}}{v_{ab}} = \frac{d_{ab}}{80}](https://tex.z-dn.net/?f=t_%7Bab%7D%20%3D%20%5Cfrac%7Bd_%7Bab%7D%7D%7Bv_%7Bab%7D%7D%20%3D%20%5Cfrac%7Bd_%7Bab%7D%7D%7B80%7D)
= distance traveled from station B to station C
= Speed of train from station B to station C = 60 kmh⁻¹
![t_{bc} = \frac{d_{bc}}{v_{bc}} = \frac{d_{bc}}{60}](https://tex.z-dn.net/?f=t_%7Bbc%7D%20%3D%20%5Cfrac%7Bd_%7Bbc%7D%7D%7Bv_%7Bbc%7D%7D%20%3D%20%5Cfrac%7Bd_%7Bbc%7D%7D%7B60%7D)
Total distance traveled is given as
![d = d_{ab} + d_{bc}](https://tex.z-dn.net/?f=d%20%3D%20d_%7Bab%7D%20%2B%20d_%7Bbc%7D)
Total time of travel is given as
![t = t_{ab} + t_{bc}](https://tex.z-dn.net/?f=t%20%3D%20t_%7Bab%7D%20%2B%20t_%7Bbc%7D)
Average speed is given as
![v_{avg} = \frac{d}{t} \\v_{avg} = \frac{d_{ab} + d_{bc}}{t_{ab} + t_{bc}}\\v_{avg} = \frac{d_{ab} + d_{bc}}{(\frac{d_{ab}}{80} ) + (\frac{d_{bc}}{60} ) }](https://tex.z-dn.net/?f=v_%7Bavg%7D%20%3D%20%5Cfrac%7Bd%7D%7Bt%7D%20%5C%5Cv_%7Bavg%7D%20%3D%20%5Cfrac%7Bd_%7Bab%7D%20%2B%20d_%7Bbc%7D%7D%7Bt_%7Bab%7D%20%2B%20t_%7Bbc%7D%7D%5C%5Cv_%7Bavg%7D%20%3D%20%5Cfrac%7Bd_%7Bab%7D%20%2B%20d_%7Bbc%7D%7D%7B%28%5Cfrac%7Bd_%7Bab%7D%7D%7B80%7D%20%29%20%2B%20%28%5Cfrac%7Bd_%7Bbc%7D%7D%7B60%7D%20%29%20%7D)
Given that :
![d_{ab} = 4 d_{bc}](https://tex.z-dn.net/?f=d_%7Bab%7D%20%3D%204%20d_%7Bbc%7D)
So
![v_{avg} = \frac{4 d_{bc} + d_{bc}}{(\frac{4 d_{bc}}{80} ) + (\frac{d_{bc}}{60} ) }\\v_{avg} = \frac{4 + 1}{(\frac{4 }{80} ) + (\frac{1}{60} ) }\\v_{avg} = 75 kmh^{-1}](https://tex.z-dn.net/?f=v_%7Bavg%7D%20%3D%20%5Cfrac%7B4%20d_%7Bbc%7D%20%2B%20d_%7Bbc%7D%7D%7B%28%5Cfrac%7B4%20d_%7Bbc%7D%7D%7B80%7D%20%29%20%2B%20%28%5Cfrac%7Bd_%7Bbc%7D%7D%7B60%7D%20%29%20%7D%5C%5Cv_%7Bavg%7D%20%3D%20%5Cfrac%7B4%20%2B%201%7D%7B%28%5Cfrac%7B4%20%7D%7B80%7D%20%29%20%2B%20%28%5Cfrac%7B1%7D%7B60%7D%20%29%20%7D%5C%5Cv_%7Bavg%7D%20%3D%2075%20kmh%5E%7B-1%7D)
2)
= Speed of train from station A to station B = 80 kmh⁻¹
= time of travel between station A to station B
= distance traveled from station A to station B
we know that
![distance = (speed) (time)](https://tex.z-dn.net/?f=distance%20%3D%20%28speed%29%20%28time%29)
![d_{ab} = v_{ab} t_{ab}\\d_{ab} = 80 t_{ab}](https://tex.z-dn.net/?f=d_%7Bab%7D%20%3D%20v_%7Bab%7D%20t_%7Bab%7D%5C%5Cd_%7Bab%7D%20%3D%2080%20t_%7Bab%7D)
= distance traveled from station B to station C
= Speed of train from station B to station C = 60 kmh⁻¹
= time of travel for train from station B to station C
we know that
![distance = (speed) (time)](https://tex.z-dn.net/?f=distance%20%3D%20%28speed%29%20%28time%29)
![d_{bc} = v_{bc} t_{bc}\\d_{bc} = 60 t_{bc}](https://tex.z-dn.net/?f=d_%7Bbc%7D%20%3D%20v_%7Bbc%7D%20t_%7Bbc%7D%5C%5Cd_%7Bbc%7D%20%3D%2060%20t_%7Bbc%7D)
Total distance traveled is given as
![d = d_{ab} + d_{bc}\\d = 80 t_{ab} + 60 t_{bc}](https://tex.z-dn.net/?f=d%20%3D%20d_%7Bab%7D%20%2B%20d_%7Bbc%7D%5C%5Cd%20%3D%2080%20t_%7Bab%7D%20%2B%2060%20t_%7Bbc%7D)
Total time of travel is given as
![t = t_{ab} + t_{bc}](https://tex.z-dn.net/?f=t%20%3D%20t_%7Bab%7D%20%2B%20t_%7Bbc%7D)
Average speed is given as
![v_{avg} = \frac{d}{t} \\v_{avg} = \frac{d_{ab} + d_{bc}}{t_{ab} + t_{bc}}\\v_{avg} = \frac{80 t_{ab} + 60 t_{bc}}{t_{ab} + t_{bc}}](https://tex.z-dn.net/?f=v_%7Bavg%7D%20%3D%20%5Cfrac%7Bd%7D%7Bt%7D%20%5C%5Cv_%7Bavg%7D%20%3D%20%5Cfrac%7Bd_%7Bab%7D%20%2B%20d_%7Bbc%7D%7D%7Bt_%7Bab%7D%20%2B%20t_%7Bbc%7D%7D%5C%5Cv_%7Bavg%7D%20%3D%20%5Cfrac%7B80%20t_%7Bab%7D%20%2B%2060%20t_%7Bbc%7D%7D%7Bt_%7Bab%7D%20%2B%20t_%7Bbc%7D%7D)
Given that :
![t_{ab} = 3 t_{bc}](https://tex.z-dn.net/?f=t_%7Bab%7D%20%3D%203%20t_%7Bbc%7D)
So
![v_{avg} = \frac{80 t_{ab} + 60 t_{bc}}{t_{ab} + t_{bc}}\\v_{avg} = \frac{80 (3) t_{bc} + 60 t_{bc}}{(3) t_{bc} + t_{bc}}\\v_{avg} = \frac{(300) t_{bc}}{(4) t_{bc}}\\v_{avg} = 75 kmh^{-1}](https://tex.z-dn.net/?f=v_%7Bavg%7D%20%3D%20%5Cfrac%7B80%20t_%7Bab%7D%20%2B%2060%20t_%7Bbc%7D%7D%7Bt_%7Bab%7D%20%2B%20t_%7Bbc%7D%7D%5C%5Cv_%7Bavg%7D%20%3D%20%5Cfrac%7B80%20%283%29%20t_%7Bbc%7D%20%2B%2060%20t_%7Bbc%7D%7D%7B%283%29%20t_%7Bbc%7D%20%2B%20t_%7Bbc%7D%7D%5C%5Cv_%7Bavg%7D%20%3D%20%5Cfrac%7B%28300%29%20t_%7Bbc%7D%7D%7B%284%29%20t_%7Bbc%7D%7D%5C%5Cv_%7Bavg%7D%20%3D%2075%20kmh%5E%7B-1%7D)