Question

A train overtakes two persons walking along a railway track. The first one walks at 4.5 km/hr. The other one walks at 5.4 km/ hr. The train needs 8.4 and 8.5 seconds respectively to overtake them. What is the speed of the train if both the persons are walking in the same direction as the train?
(a) 63 km/hr
(b) 72 km/hr
(c) 99 km/hr
(d) 81 km/hr

Answer 1

Hint: To solve this problem, we should know the basics of speed, distance and time. Further, the formulas in concern are $\text{Speed = }\dfrac{Dis\tan ce}{time}$. Further, we will also use the conversion of hours into seconds, which is given by 1 hour = 3600 seconds. We will use these formulas to find the speed of the train.

Complete step-by-step answer:

Before beginning to solve the problem, we should first try to convert the word problem in hand into the relevant mathematical equations and convert the time units into hours (1 second = $\dfrac{1}{3600}$ hours). We start with the first condition in the problem which states that the train needs 8.4 seconds to overtake the person with speed 4.5 km/hr. Let the distance between the train and both the people be x. Let the speed of the train be v km/hr. Thus, in 8.4 seconds $\left( =\dfrac{8.4}{3600}\text{ hours} \right)$, the distances travelled by the formula – Distance = Speed $\times $ time.
For, train, we have,
Distance = $\dfrac{8.4}{3600}$ v
For, the first person, we have,
Distance = $4.5\left( \dfrac{8.4}{3600} \right)$ = 0.0105 km (converting seconds to hours..
Thus, since the distance between train and the person is x, we have,
$\dfrac{8.4}{3600}$ v = 0.0105 + x -- (1)
We then start with the second condition in the problem which states that the train needs 8.5 seconds to overtake the person with speed 5.4 km/hr. Thus, in 8.5 seconds $\left( =\dfrac{8.5}{3600}\text{ hours} \right)$, the distances travelled by the formula – Distance = Speed $\times $ time.
For, train, we have,
Distance = $\dfrac{8.5}{3600}$ v
For, the first person, we have,
Distance = $5.4\left( \dfrac{8.5}{3600} \right)$ = 0.01275 km (converting seconds to hours..
Thus, since the distance between train and the person is x, we have,
$\dfrac{8.5}{3600}$ v = 0.01275 + x -- (2)
Solving (1) and (2), we get,
v = 81 km/hr
Hence, the correct answer is (d) 81 km/hr.

Note: Another alternative way to solve this problem is by using the concept of relative motion. In this case, we calculate the relative speed which is (v – 4.5) km/hr and (v-5.4) km/hr. Now, relative distance is x km (as calculated in the problem). Thus, using Distance = Speed $\times $ time, we have,
x = (v-4.5) $\times $ $\dfrac{8.4}{3600}$ = (v-5.4) $\times $ $\dfrac{8.5}{3600}$ (here, since, we use relative motion, we use relative speed)
Also, x is used since we have assumed that the original distance between the train and the people is x km.
Solving, we will get v = 81 km/hr (same as the above solution).