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# Time, Speed and Distance - Introduction

Time and distance as a topic involves a variety of areas which include speed-time-distance concepts, relative speed, moving and stationary bodies, boats and streams, circular motion, and so on. While the diversity of problems from this area is vast, the concepts are not many and once grasped, will enable you to solve many of the problems with ease. A familiarity with the types of problems will also help.

### Types of Problems

The following table gives the various types of problems and the approach used for each one. A word of caution here: while it is a good idea to have some approaches in hand while attempting problems, it is very important to analyse each problem on its own merit and then decide the exact approach required. Blindly following the approaches given below could result in wrong answers.

 Type of problem Approach Approaching or receding bodies – two trains approach each other and pass by, what is the time taken for passing? Use concept of relative speed and the basic idea of “speed = distance/ time” to form an equation that can be solved for time Boat traveling with or against the cur- rent – what is the effective speed, or what is the time required to reach a particular point up/downstream? Use the concept of boats and streams (see later) to form suitable equations that can be solved for the required unknown Races along a straight track – how much start should the faster runner give the slower one so that they both finish together? Decide which quantity (distance or time) is the same for both runner and equate its formula on both sides, then solve for the unknown Races along a circular track – when is the first meeting, when is the first meeting at the starting point? Understand the problem thoroughly and decide which of the formulas to apply Other problems on time and distance Understand the problem thoroughly, express the given data in equations as far as possible, and decide which of the basic formulae to apply

### Speed

The concept of speed is basic to solving problems involving relative motion, circular motion, etc. Assume that a distance ‘d’ is covered in a time duration ‘t’. Then, the average speed ‘s’ or simply the speed is defined as the rate of covering the given distance and is given by

s = d/t.

Relations between speed, distance and time

From the above, the following can be concluded about the relations between speed, distance and time.

If two bodies move at the same speed, the distances covered by them are (directly) proportional to the times of travel. i.e., when s is constant,

If two bodies move for the same time period, the distances covered by them are (directly) proportional to the speeds of travel. i.e., when t is constant,

If two bodies move for the same distance, their times of travel are inversely proportional to the speeds of travel. i.e., when d is constant,

### Units of speed and conversion factors

The units of speed are kilometre per hour (kmph or km/h) and metre per second (m/s).

To convert a speed given in m/s into a speed in km/h, multiply with 18/5. To convert a speed given in km/h into a speed in m/s, multiply with 5/18.

A simple way to remember the multiplying factor is to recall that a particular speed when expressed in km/h is numerically larger than the same speed expressed in m/s.

For other units like m/min or km/min, it is sufficient to remember that 1 km = 1000m and 1 min = 60 sec.

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