Question

On a 120 km track, a train travels the first 30 km with a uniform speed of 30 km/h. How fast must the train travel the next 90 km so as to average 60 km/h for the entire trip?
A. 65 km/h
B. 90 km/h
C. 120 km/h
D. 100km/h

Answer 1

Hint: Use the equation \[{{v}_{av}}=\dfrac{d}{t}\] to find the average time. Then find the time taken for the second part using \[t={{t}_{1}}+{{t}_{2}}\]. And thus find \[{{v}_{2}}\] systematically. Or use \[{{v}_{av}}=\dfrac{{{v}_{1}}+{{v}_{2}}}{2}\].

Complete Step-by-Step solution:
In order to understand let us draw a simple representation using the given data.

From the above representation,
The total distance travelled by the train, d = 120km
The average velocity of the entire trip,\[{{v}_{av}}\]=60km/h
Initial distance travelled, \[{{d}_{1}}\]=30 km
Velocity at which it travelled the initial 30 km, \[{{v}_{1}}\] =30km/h
Remaining distance travelled, \[{{d}_{2}}=90km\]
What we have to find is
t = total time taken
\[{{t}_{1}}\] = time taken to travel the first 30km
\[{{t}_{2}}\]= time taken to travel the next 90 km
\[{{v}_{2}}\]= the velocity at which the train covered the remaining 90 km.
Let us start our calculation.
In order to calculate the velocity at the second interval of time we need to first find the total time taken by the train to finish the journey.
We know the equation to find the average velocity. i.e.
\[Average\text{ }velocity=\dfrac{Total\text{ }distance\text{ }travelled~~}{Total\text{ }time\text{ }taken}\] or
\[{{v}_{av}}=\dfrac{d}{t}\]…………(1)
Using eq (1), we can calculate the value of t. i.e.
\[t=\dfrac{d}{{{v}_{av}}}\]
\[t=\dfrac{120km}{60km/h}\]
t = 2h
similarly,
\[{{t}_{1}}=\dfrac{{{d}_{1}}}{{{v}_{1}}}\]
\[{{t}_{1}}=\dfrac{30km}{30km/h}\]
\[{{t}_{1}}=1h\]
Then \[{{t}_{2}}=t-{{t}_{1}}\]………..(2)
By substituting the values of t and\[{{t}_{1}}\] in equation (2), we get the value of \[{{t}_{2}}\] as
\[{{t}_{2}}=t-{{t}_{1}}=2-1\]
\[{{t}_{2}}=1h\]
Now we can find the velocity at which the train travelled the remaining 90 km is
\[{{v}_{2}}=\dfrac{{{d}_{2}}}{{{t}_{2}}}\]…………….(3)
\[{{v}_{2}}=\dfrac{90km}{1h}\]
\[{{v}_{2}}=90km/h\]

With a velocity of 90 km/h the train covered the remaining 90 km in one hour.
Thus option B is the correct answer.

Note: Always try to use the simple method to solve the problem. Please keep in mind that while finding the velocities; always remember to substitute the positive and negative signs since you are dealing with vector quantities.