The concept of Relative Speed is central to most physics. When do we say that a body is in motion? If you are lying perfectly still on the ground, are you in motion or are you at rest? The answer is that if you change your relative position with respect to something, you are in motion. Otherwise, you are in a state of rest.

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## Relative Speed

Imagine that you have a dart gun with a muzzle velocity of 45 mph. Further imagine that you are on a bus travelling along a straight highway at 55 mph and that you point the gun so that the barrel is level and pointing directly forward, toward the front of the bus. Assuming no recoil, as it leaves the muzzle of the gun, how fast is the dart travelling relative to the road? That’s right!

100 mph. The dart is already travelling forward at 55 mph relative to the road just because it is on a bus that is moving at 55 mph relative to the road. Add to that the velocity of 45 mph that it acquires as a result of the firing of the gun and you get the total velocity of the dart relative to the road.

In other words, we say that the velocity of one object as measured from another object is its Relative Speed with respect to it. There are broadly two cases in the relative speed as shown below.

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## When the Objects are moving in the same Direction

Let us say two objects with a velocity equal to a and b with respect to the ground are moving in the same direction. Suppose you and your friend are both travelling in a train or you are on a train and she is in a car but both are travelling towards the same destination. Let your velocity be a and that of your friend be b. Then the Relative Speed of your friend as seen by you will be equal to:

a – b. This is known as the Relative Speed of a with respect to b.

## When the Objects are moving in the Opposite Direction

Let us say you are going from Bombay to Pune and your friend is travelling from Pune to Bombay in opposite directions. Then we can say that the Relative Speed is as: (a + b) or the sum of the two velocities.

Example 1: A bus is travelling along a straight highway at a constant 55 mph. A person sitting at rest on the bus fires a dart gun that has a muzzle velocity of 45 mph straight backwards, (toward the back of the bus). Find the velocity of the dart, relative to the road, as it leaves the gun.

Answer: Defining: v_{BR }to be the velocity of the bus relative to the road, v_{DB }to be the velocity of the dart relative to the bus, and v_{DR }to be the velocity of the dart relative to the road, and defining the forward direction to be the positive direction; we have:

v_{DR }= v_{BR }− v_{DB }or in other words we have: v_{DR} = 55 mph − 45 mph. Therefore we may write that v_{DR }= 10 mph in the direction in which the bus is travelling.

## Conversions

While solving any problem on the concept of speed, the relative units have to be uniform. In other words, the units throughout any calculation shall be the same. The most common units are the meter and the Kilometer. The following is the conversion between the units of distance.

### Distance Conversions

1 m is equivalent to 1.0936 yards, or 39.370 inches. Also, we have 1/100 m = one centimetre 1/1,000 m = one millimetre.

We can easily make the conversions from one system of units, for example, a man covers 30 ft in ten seconds. What is his speed in meter per second?

A) 0.9146 m/s. B) 9.146 m/s C) 13.32 m/s D) 1.332 m/s

Answer: We know that 1m is equal to 3.28 ft. So we can write that 30 ft = (1/3.28)×30 = 9.146 m. So the speed of the man is 9.146/10 m/s or 0.9146 m/s. Therefore the correct option is D) 1.332 m/s.

### Time Conversions

The time conversions are simple. We know that each minute has 60 seconds and each hour has 60 minutes. Therefore the number of seconds in an hour = 60×60 = 3600 s.

## Trick For The Speed Conversion

Now let us use both the tricks to do the most common conversion that you will encounter in the section. The most common conversion is from km/hr to m/s. Let us see what we can do here. Convert one kilometre per hour to meter per second?

Answer: We know that 1 km = 1000 m. Also as we just saw 1 hr = 3600 s.

So we can write 1km/1hr = 1000/3600 m/s = 5/18 m/s.

For example, a body is moving at a velocity of 60 km/hr. Its speed in m/s is?

Answer: If you want to change any speed from km/hr to m/s, you need to multiply the km/hr by the factor 5/18 as we just saw. So, in this case, we can write:

60 km = 60×(5/18) m/s = 50/3 m/s = 16.67 m/s.

## Practice Question:

Q 1: A train is travelling from a station A to another station B at a speed of 60 km/hr. Another train is travelling at a speed of 30 km/hr from station B to A. The relative speed of train A with respect to B is?

A) 90 m/s B) 30 m/s C) 25 m/s D) 12 m/s

Ans: C) 25 m/s