Time and distance are two quantities which are dependent on speed as well. Whenever we talk about time and distance we cannot neglect the idea of speed.

## Distance and Time Definition

Distance is the path that is covered by a moving object which can be a person, a vehicle, or any other thing that can move. Speed is the rate at which the object is moving for example 20m/s means the object is moving at the rate of 20 m per second which more precisely means the object is covering a distance of 20 meters in one second.

Distance traveled is directly proportional to speed as well as time. This implies, if either speed or time or both increase so will the distance will increase accordingly and if either or both decrease so will the distance will. Also, time and speed are inversely proportional to each other, that is, if speed is increased, time decreases and is speed decreases time taken increases.

## Distance and Time Formula

The formula that is used to evaluate distance, speed or time when the other two are known is:

$Distance\ =\ Speed\ \times\ Time$

That is, distance traveled can be calculated by determining the product of the speed of the object and the time taken by it.
From this we can evaluate,

$Speed$ = $\frac{Distance}{Time}$

$Time$ = $\frac{Distance}{Speed}$

## How to Find Distance and Time

For determining any value from the above formulas it is important that the given values should be same format. For example if we are given distance in kilometers and speed is given in m/s (meter per second), then we must either convert speed to km/h (kilometers per hour) or convert distance in meters. Units must always be same for any computation to be performed.

When we want to convert distance we can use the following conversions

1 km = 1000m

1 m = 100cm

For converting speed we can use the following conversions

1 km/h = $\frac{5}{18}$ m/s

For converting time we can use the given conversions

1 hr = 60 min

1 min = 60 s

1 hr =3600 s

## Distance and Time Graph

The slope of the curve of distance and time graph gives the value of the speed.

This is a simple example for distance vs time graph. But it is not necessary to have a straight line graph every time. The straight line graph of distance vs time implies that the object is moving with a constant speed. When we have any other curve of distance time graph this implies that the speed of the object is varying from time to time.

This graph shows that the speed is changing very quickly.

In any of the above cases it is to be noted that whether speed is constant or changing, the distance always increases with time.

## Distance and Time Problems

Let us see some examples based on distance and time for a better understanding.
Example 1:

What is the time taken by a man to take a run up of 100 m at the speed of 36 km/h?

Solution:

The distance is given to be 100 m.

The speed of the man is 36 km/h = $(36\ \times$ $\frac{5}{18}$$)$ m/s = 10 m/s

Therefore time taken for the given run up = $\frac{distance}{speed}$ = $\frac{100}{10}$ = 10 s

The man will take 10 seconds for the run up of 100 m at the speed of 36 km/h.
Example 2:

A train is passes a pole in 9 seconds completely and the same train passes the platform in 36 seconds completely. If the length of the platform is given to be 780 m, then find the length of the train.

Solution:

Let the length of the train be $x\ m$.

Distance covered by train in passing the platform = length of train + length of platform

$\Rightarrow$ Distance covered in passing platform = $x\ +\ 780$

Now according to question we have,

Speed of the train = $\frac{x}{9}$ m/s

Also, speed of the train = $\frac{(x + 780)}{36}$ m/s

Therefore, $\frac{x}{9}$ = $\frac{(x\ +\ 780)}{36}$

$\Rightarrow\ 36\ x\ =\ 9\ (x\ +\ 780)$

$\Rightarrow\ 4\ x\ =\ x\ +\ 780$

$\Rightarrow\ 3\ x\ =\ 780$

$\Rightarrow\ x\ =\ 260$

Therefore, the length of the train is 260 meters.
Example 3:

The speed of the boat in still water is 10 km/h. A man is taking four times the time to row up the boat in river as to row down the boat in river. Find the rate of the stream.

Solution:

Let the rate of stream be $x$ km/h

Then the speed of boat downstream = $(x\ +\ 10)$ km/h

And the speed of boat upstream = $(10\ –\ x)$ km/h

According to the question,

$x\ +\ 10\ =\ 4\ (10\ –\ x)$

$\Rightarrow\ x\ +\ 10\ =\ 40\ –\ 4x$

$\Rightarrow\ 5x\ =\ 30$

$\Rightarrow\ x\ =\ 6$

Therefore, the rate of the stream is 6 km/h.