Maths

Time and Speed

By Praveen Dwivedi -

November 11, 2010

9451

Some principles which are used while solving questions related to trains and cars.

The time taken to pass a given object

A train x metres long, while crossing a point object ( like a man, pole or tree) takes the same time as it would to cross the distance of x metres.
If the train ( of length x metres) has to cross a platform or bridge of length y metres the time taken will be same as the time taken to cover the distance of ( x+y) metres.

Relative speed

If two vehicles of speeds u km/hr and v km/hr are moving in the opposite direction, their relative speed is (u+v) km/hr
If two vehicles of speeds u km/hr and v km/hr are moving in the same direction, their relative speed is (u-v) km/hr, where u>v

The time taken to overtake

If two trains of length x km and y km are moving in the same direction at speeds of u km/hr and v km/hr, the time taken by the faster train to completely overtake the slower train is (x+y)/(u-v) km/hr where u km/hr is the speed of the faster train.
If two trains of length x km and y km are moving in the opposite directions at speeds of u km/hr and v km/hr, the time taken by them to completely pass each other is (x+y)/(u+v) km/hr.

Boats and ships- the concept of ‘upstream’ and ‘downstream’

When boats travel, their speed is influenced by the flow of the water. When a boat moves against the current, it is said to be upstream. When it moves along with the current it is said to be downstream. If the speed of the boat is x km/hr and the speed of the current is y km/hr:

While moving upstream the actual speed will be (x-y) km/hr
While moving downstream the actual speed will be (x+y) km/hr

The time taken by the boat to go downstream is always less than the time taken by it to go upstream.

The following examples will illustrate these formulae and principles

1. How much time does a train 150 metre long, travelling at a speed of 10 m/sec take to cross a 100 metre long platform?

Total Distance = 150m + 100m = 250m

Speed = 10m/sec

Using

We get time = 250/10 sec = 25 seconds

2. If a car is travelling at a speed of 40 km/hr and a truck travelling at a speed of 20 km/hr in the opposite direction crosses it. What is the relative speed of the truck as felt by a passenger of the car?

Since they are travelling in opposite directions, the relative speed will be (u+v) km/hr,

i.e. 20 + 40 km/hr

= 60 km/hr

3. How much time will a train 100 m long, travelling at a speed of 180 km/hr take to overtake a train 200 m long travelling at a speed of 90 km/hr?

Time taken = ( total distance) / relative speed

Total distance = 100 + 200 = 300 m

Relative speed = 180- 90 = 90 km/hr = 25 m/sec

Therefore time taken= 300m/25m/sec = 12 seconds

4. The speed of a boat in still water is 11 km/hr. It takes 4 hours for it to cover 56 km while going downstream. In how much time will it cover the same distance while travelling upstream?

Using

We can calculate the speed of the boat while moving downstream is 56 km/ 4 hours= 14 km/hr

If the speed of the boat in still water is x km/hr, ( 11 km/hr, given)

And the speed of the stream is y km/hr, ( to be calculated)

Then the speed while moving downstream = (x+y) km/hr ( 14 km/hr as calculated)

Substituting the values we get

14 km/hr= 11km/hr + y km/hr
Y= 3, i.e. the speed of the stream is 3 km/hr

Now, the speed of the boat while travelling upstream = (x-y) km/hr

(11-3) km/hr = 8 km/hr

Using

We get the time= 56 km/ 8 km/hr = 7 hours

5. A man covers a distance of 20 kilometres at a speed of 60 km/hr and went back to the starting point at a speed of 40 km/hr. What is the average speed over the round trip?

Average speed = total distance/ total time

Total distance = 20 + 20 = 40 km

Total time = time taken in first case + time taken in second case

Time taken in first case = distance/ speed = 20/60 = 1/3 hours

Time taken in second case = distance/ speed = 20/40 = 1/2 hours

Total time = 1/3 + 1/2 = 5/6 hours

Average speed= 40/5/6 = 48 km/hr

The following questions deal with the principles given above.

If a car covers 162 kilometres in one hour, how many metres can it cover in one second?
A man cycles to work. He covers two-third of the distance at the speed of 20km/hr and the rest of the distance at a speed of 10 km/hr. What is the overall speed?
A man reaches the post office 30 minutes after leaving from home. He cycles at a speed of 12 km/hr. As soon as he reaches the post office he immediately starts walking back home. If he takes ninety minutes to reach home, what is his speed?
A train which is 750 m long takes 70 seconds to pass a platform which is one kilometre long. Find the speed of the train in km/hr.
A person can row a distance of one kilometre downstream in 15 minutes and return in 30 minutes. What is the velocity of the current?
If a man walks to his office at a speed of 4 km/hr, he is late by 5 minutes. If he walks at a speed of 5 km/hr, he reaches 2.5 minutes early. Find the distance of his office from his home.
A train which is 100 metres long takes 7.2 seconds to cross a man walking at a speed of 5 km/hr in the opposite direction. Find the speed of the train.
Two trains are travelling in the same direction on parallel tracks. The length of the trains is 750m and 1050m and the speeds are 102 and 72 km/hr respectively. How many seconds will be required for them to completely clear each other?
Two trains travelling in opposite directions at the same speed cross each other in 10 seconds. The length of one train is 150m and the length of the other is 200m. Find the speed of the trains.
There are two places Shantown and Mayle which are located 162 kilometres apart. Two trains start from the two towns at the same time and meet each other after six hours. If the speed of the train moving from Mayle to Shantown is 8 km/hour than the train moving in the opposite direction, find the speeds of the two trains.
A train has 24 carriages, the length of each being 60 metres. The length of the engine is also 60 metres. The train is travelling at a speed of 60 km/hr. Find the time it will take to cross a bridge which is 1.5 kilometres long.
If the speed of a train is 72 km/hr it takes 18 seconds to completely pass a man standing near the tracks. If the train maintains the same speed, it takes 30 seconds to cross a platform. Find the length of the platform.
There is a cycling track which is 45 kilometres long. Two cyclists start at the same time but from the two different ends of the track. If the speed of one is 16 km/hr and the speed of the other is 14 km/hr, after how much time will they meet?
A boat takes four hours to cover a certain distance along the direction of the flow of the river. The speed of the river is 3 km/hr. The boat takes five hours to make the return journey. What is the speed of the boat?
There are two parallel tracks between two towns Fairisle and Springland. One train which is 150 m in length, moves at a speed of 60 km/hr. Another train, 120 m in length, leaves 5 minutes after the first train in the same direction and has a speed of 90km/hr. At what distance will the second train completely overtake the first train?