Time, Speed and Distance
Time, speed and distance is the chapter that measures how fast a journey is made, and it is one of the most regularly asked topics in the ADRE Grade IV, APSC and Assam Police papers. The whole chapter grows from a single relation: speed is the distance covered divided by the time taken. From this one line the candidate can find the distance when the speed and the time are known, and the time when the speed and the distance are known. The method followed in the Master Book: Maths Masti (D. Hurina) is to write the three quantities in a small triangle, distance on top and speed and time below, and to cover the quantity that is sought, so that the correct formula appears at once. Almost every question in the chapter is a variation of this first idea, or of the special cases of trains and boats built upon it.
Basic Terms
- Speed: the distance covered in unit time.
- Distance: the length of the path travelled.
- Time: the duration of the journey.
- Average speed: the total distance divided by the total time.
- Relative speed: the speed of one body with respect to another.
- Uniform speed: equal distances covered in equal intervals of time.
The three quantities are locked together, and any two of them fix the third. The trick of every question is only to decide which two are given and which one is asked.
The Fundamental Relation and Its Forms
Speed equals distance divided by time. Distance equals speed multiplied by time. Time equals distance divided by speed. These three forms are the same relation written three ways, and the candidate should be able to move among them without hesitation. If any two of the three quantities are known, the third follows immediately.
Units and Conversions
Speed is commonly given in kilometres per hour or in metres per second. To change kilometres per hour into metres per second, multiply by five and divide by eighteen. To change metres per second into kilometres per hour, multiply by eighteen and divide by five. Time is often given in minutes and seconds, which must be written as a fraction of an hour or of a minute before the formula is used. A wrong conversion is the most common source of error in this chapter.
Average Speed
Average speed is the total distance divided by the total time, and it is not the average of the several speeds. If a journey is made in two equal distances at speeds x and y, the average speed is twice x y divided by x plus y. When the distances are not equal, the safest method is to find the total distance and the total time separately and then divide.
Relative Speed
When two bodies move in the same direction, the relative speed is the difference of their speeds. When they move in opposite directions, the relative speed is the sum of their speeds. This one idea solves all questions of two trains, two walkers or two vehicles that approach or overtake each other.
Problems on Trains
A train passing a pole or a man covers its own length. A train passing a platform, a bridge or another train covers the sum of the lengths involved. When two trains move in the same direction, the relative speed is the difference; when they move in opposite directions, it is the sum. The candidate must read carefully what the train is passing, because that decides the distance to be used.
Boats and Streams
When a boat moves with the stream, its speed is the speed of the boat in still water plus the speed of the stream, and against the stream it is the difference. These two results also give the speed of the boat in still water as half the sum of the two speeds, and the speed of the stream as half the difference.
How This Chapter Appears in the Examination
In ADRE Grade IV, APSC and Assam Police papers the questions usually ask for the distance covered at a given speed and time, the time taken for a given journey, the average speed over a journey in several parts, the time for two trains to cross, or the speed of a boat and stream. The numbers are kept simple and the relation is always the same, so accuracy in conversion and careful reading are what earn the mark.
Shortcut Methods for Rapid Calculation
Learn the five-eighteenth factor by heart, since it converts kilometres per hour to metres per second. When a distance is covered at two speeds for the same distance, use the harmonic formula for average speed. For crossing questions write the total length first, then apply the relative speed. Cancel the units before multiplying, so that the arithmetic stays small.
Common Mistakes to Avoid
- Taking the average of the speeds instead of the total distance divided by the total time.
- Using kilometres and metres together without converting.
- Forgetting to add the length of the platform or the other train.
- Adding the speeds when the bodies move in the same direction.
- Mixing up the with-stream and against-stream cases.
How to Prepare This Topic
- Fix the three forms of the speed relation in memory.
- Practise kilometre-per-hour and metre-per-second conversions daily.
- Work on average speed with equal and unequal distances.
- Practise train-crossing and boat-stream questions as separate sets.
- Finish with a mixed set and revise the single relation every week.
Frequently Asked Questions
Why is average speed not the mean of the speeds?
Because the time spent at each speed may differ. Average speed is the total distance divided by the total time, which is a weighted figure and not the simple mid-value of the speeds.
How do I convert kilometres per hour to metres per second?
Multiply by five and divide by eighteen. For example, thirty-six kilometres per hour is ten metres per second. To reverse the change, multiply by eighteen and divide by five.
What distance does a train cover while passing a platform?
The length of the train plus the length of the platform. While passing a pole or a standing man the train covers only its own length.
How do I handle two bodies moving together?
Use the relative speed. When they move in the same direction, subtract the speeds; when they move in opposite directions, add them. Divide the distance between them by the relative speed to get the time.
How do boat and stream speeds combine?
With the stream the speeds add, and against the stream they subtract. The boat's speed in still water is half the sum of the two, and the stream's speed is half the difference.