In this article, we explore the concept of Average Speed.

Average speed is simply the total distance travelled divided by the total time taken. Example:

Aragorn, your watchman, took part in a two-stage race where he travelled by car and on foot. His speed in the two stages was 60 km/hr and 15 km/hr respectively. In each stage, he covered 60 kms. Calculate Aragorn’s average speed across the entire race.

In this question, one will be tempted to simply take the average of the two speeds given and calculate the overall average speed as (60 + 15)/2 = 37.5 km/hr.

But, this is not the actual average speed.

Remember: Average speed = Total distance travelled / Total time taken

In this case, time taken in the first stage = 60/ 60 = 1 hr

Time taken in the second stage = 60/ 15 = 4 hrs

Hence, total time taken = 4 + 1 hrs = 5 hrs

Total distance covered = 60 + 60 = 120 kms

Hence, average speed = 120/ 5 = 24 km/hr

Note: If a journey involves two stages where the speeds in the two stages are x & y but the distance covered in the two stages is the same then the average speed = 2xy/(x + y)

Hence, Aragorn’s average speed

= (2*60*15)/(60 + 15)

= 1800/75 = 24 km/hr

So the above formula is applicable when the distance covered in the stages is equal. But is there a formula to calculate average speed if the time taken in each stage is equal?

When the time taken for each of the stages of a journey is the same, you can directly take the average of the individual speeds. This average would be the overall average speed.

Example:

A bus travelled at a speed of 30 km/hr for 2 hours on a busy road. It then reached a highway and travelled at a speed of 60 km/hr for another 2 hours. Calculate the average speed of the bus?

Explanation:

Since the time taken in each stage is the same, we simply find the average of the two speeds. Hence, overall average speed = (30 + 60)/ 2 = 45 km/hr

Let us now discuss a few easy problems involving trains.

Note: When we discuss cars, bikes, etc. overtaking each other, we do not consider their lengths. But, in the case of trains, we must consider their lengths.

A Rajdhani train with a length of 200 metres crosses a station in just 30 seconds. If the speed of the train was 90 km/hr, what was the length of the station platform?

Explanation: We have already seen that we need to consider the length of the train. Similarly, we must also consider the length of platforms, etc.

In this question, speed of the train = 90 * 5/18 = 25 m/s.

Distance covered by the train in 30 seconds = 30 * 25 = 750 m

This 750 metres must be the sum of the length of the platform and the length of the train.

Hence, length of the platform = distance covered – length of the train = 750 – 200 = 550 metres

Let us consider another interesting problem:

A Japanese Maglev train travelling at 380 km/hr takes 15 seconds to pass a man riding his Suzuki GSX bike in the same direction at a speed of 272 km/hr. Find the length of the train.

Explanation:

Here this problem involves relative speed. Since the train and the bike are going in the same direction, we must take the difference of the two speeds to get the relative speed.

Hence, relative speed = 380 – 272 = 108 km/hr

Since the time given is in seconds, let us convert this speed to m/s.

Hence, relative speed = 108 * 5/18 = 30 m/s

As the train takes 15 seconds to pass the bike, the distance covered in this time would be the length of the train.

Therefore, length of the train = distance covered = time * speed
= 15 * 30 = 450 metres