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Studying Physics Topics can lead to exciting new discoveries and technological advancements.

## Explain Why Speed and Velocity are Not Always Equal in Magnitude

The speed of a car (or any other body) gives us an idea of how fast the car is moving but it does not tell us the direction in which the car is moving. Thus, to know the exact position of a moving car we should also know the direction in which the car is moving.

In other words, we should know the speed of the car as well as the direction of speed. This gives us another term known as Velocity which can be defined as follows : Velocity of a body is the distance travelled by it per unit time in a given direction. That is :

Velocity = \(\frac{\text { Distance travelled in a given direction }}{\text { Time taken }}\)

If a body travels a distance ‘s’ in time T in a given direction, then its velocity ‘v’ is given by :

v = \(\frac{s}{t}\)

v = velocity of the body

s = distance travelled (in the given direction)

and t = time taken (to travel that distance)

We know that the ‘Distance travelled in a given direction’ is known as ‘Displacement’. So, we can also write the definition of velocity in terms of ‘Displacement’. We can now say that: Velocity of a body is the displacement produced per unit time.

We can obtain the velocity of a body by dividing the ‘Displacement’ by ‘Time taken’ for the displacement. Thus, we can write another formula for velocity as follows :

Velocity = \(\frac{\text { Displacement }}{\text { Time taken }}\)

v = \(\frac{s}{t}\)

v = velocity of the body

s = displacement of the body

and t = time taken (for displacement)

The SI unit of velocity is the same as that of speed, namely, metres per second (m/s or m s^{-1}). We can use the bigger unit of kilometres per hour to express the bigger values of velocities and centimetres per second to express the small values of velocities. It should be noted that both, speed as well as velocity, 1 are represented by the same symbol v.

The difference between speed and velocity is that speed has only magnitude (or size), it has no specific direction, but velocity has magnitude as well as direction. In fact, velocity of a body is its speed in a specified direction (in a single straight line). Speed is a scalar quantity (because it has magnitude only).

Velocity is a vector quantity (because it has magnitude as well as direction). For example, the expression ’25 km per hour7 is the speed (because it has magnitude only), but the expression 25 km per hour towards North (or any other direction) is velocity (because it has both magnitude as well as a specified direction).

To be strictly accurate, whenever velocity is expressed, it should be given as speed in a ‘certain direction’. Usually, however, velocities are expressed without mentioning direction for the sake of convenience. The direction is assumed without being stated. The direction of velocity is the same as the direction of displacement of the body.

The ‘Distance travelled’ by a body in a given direction divided by ‘Time’ gives us average velocity. For example, if a car travels a distance of 100 km in 4 hours in the North direction, then its average velocity is \(\frac{100}{4}\) = 25 km per hour, due North.

We have just seen that, v = \(\frac{s}{t}\)

So, s = v × t

Thus, Distance travelled = Average velocity × Time

This formula should be memorized because it will be used in solving numerical problems.

### Uniform Velocity (or Constant Velocity)

If an object travels in a specified direction in a straight line and moves the same distance every second, we say that its velocity is uniform. Thus, A body has a uniform velocity if it travels in a specified direction in a straight line and moves over equal distances in equal intervals of time, no matter how small these time intervals may be.

The velocity of a body can be changed in two ways :

- by changing the speed of the body, and
- by keeping the speed constant but by changing the direction.

When a body does not cover equal distances in equal intervals of time, the velocity is said to be variable or non-uniform velocity. In this case the speed of the body is not constant. Even if the speed of a body is constant but the direction is changing, the velocity will not be uniform.

Suppose a car is moving on a circular road with constant speed. Now, though the speed of the car is constant, its velocity is not constant because the direction of car is changing continuously.

### Speed and Velocity are Not Always Equal in Magnitude

In most of the cases, the magnitude of speed and velocity of a moving body is equal. This will become clear from the following example. Suppose a boy runs a distance of 100 metres in 50 seconds in going from his home to a shop in the East direction in a straight line (see Figure).

Here, Speed of boy = \(\frac{\text { Distance travelled }}{\text { Time taken }}\)

= \(\frac{100 \mathrm{~m}}{50 \mathrm{~s}}\)

= 2 m/s ……………. (1)

Since the boy runs in a specified direction (East) in a straight line path, therefore, the displacement here will be equal in magnitude to the distance travelled. The displacement will actually be 100 m towards East. Thus,

Velocity of boy = \(\frac{\text { Displacement }}{\text { Time taken }}\)

= \(\frac{100 \mathrm{~m} \text { towards East }}{50 \mathrm{~s}}\)

= 2 m/s towards East

We can see that in this case the magnitude (or size) of speed and velocity of the boy is equal (being 2 m/s).

Please note that the magnitude of speed and velocity of a moving body is equal only if the body moves in a single straight line (like the boy in the above case). If, however, a body does not move in a single straight line, then the speed and velocity of the body are not equal. This will become clear from the following example.

Suppose the boy first runs a distance of 100 metres in 50 seconds in going from his home to the shop in the East direction, and then runs a distance of 100 metres again in 50 seconds in the reverse direction from the shop to reach back home from where he started (see Figure).

In this case, the total distance travelled is 100 m + 100 m = 200 m and the total time taken is 50 s + 50 s = 100 s. Thus,

Speed of boy = \(\frac{\text { Distance travelled }}{\text { Time taken }}\)

= \(\frac{200 \mathrm{~m}}{100 \mathrm{~s}}\)

= 2 m/s …………….. (3)

The boy runs 100 m towards East and then 100 m in opposite direction (towards West), and reaches at the starting point, his home. So, the displacement (distance travelled in a given direction) will be 100 m – 100 m = 0 m. The total time taken is 50 s + 50 s = 100 s. In this case :

Velocity of boy = \(\frac{\text { Displacement }}{\text { Time taken }}\)

= \(\frac{0 \mathrm{~m}}{100 \mathrm{~s}}\)

= 0 m/s …………. (4)

This means that when a boy first runs a distance of 100 metres towards East in 50 seconds and then changes direction and runs the same distance of 100 metres in the reverse direction (towards West) in 50 seconds, then his average speed is 2m/s but his average velocity is 0 m/s.

So, in this case the magnitude of speed and velocity of the boy is not equal. This is an unusual situation. It has happened because the boy has not moved in a single straight line. He has changed (or rather reversed) his direction of motion after reaching the shop.

In most of the cases, the bodies (or objects) move in single straight line (without changing direction). The values of speed and velocity will be the same in these cases. The difference in the values of speed and velocity arises only when a body (or object) does not move in a single straight line (and changes its direction of motion at some point of time).

From this discussion we conclude that though the average speed of a moving body can never be zero, but the average velocity of a moving body can be zero. Let us solve some problems now.

**Example Problem 1.**

A car travels a distance of 200 km from Delhi to Ambala towards North in 5 hours. Calculate (i) speed, and (ii) velocity, of the car for this journey.

**Solution:**

(i) Speed = \(\frac{\text { Distance travelled }}{\text { Time taken }}\)

= \(\frac{200 \mathrm{~km}}{5 \mathrm{~h}}\)

= 40 km/h

Thus, the speed (or average speed) of the car is 40 km/h.

(ii) Velocity = \(\frac{\text { Displacement }}{\text { Time taken }}\)

= \(\frac{200 \mathrm{~km} \text { towards North }}{5 \mathrm{~h}}\)

= 40 km/h towards North

So, the velocity (or average velocity) of the car is 40 km/h towards North.

**Example Problem 2.**

A bus covers a distance of 250 km from Delhi to Jaipur towards West in 5 hours in the morning and returns to Delhi in the evening covering the same distance of 250 km in the same time of 5 hours. Find (a) average speed, and (b) average velocity, of the bus for the whole journey.

**Solution:**

(i) Average Speed = \(\frac{\text { Total distance travelled }}{\text { Total time taken }}\)

= \(\frac{250 \mathrm{~km}+250 \mathrm{~km}}{5 \mathrm{~h}+5 \mathrm{~h}}\)

= \(\frac{500 \mathrm{~km}}{10 \mathrm{~h}}\) = 50 km / h

Thus, the average speed of the bus for the whole journey (both ways) is 50 kilometres per hour.

(ii) In this case, the bus travels 250 km from Delhi to Jaipur towards West and then comes back to starting point Delhi in the reverse direction. So, the total displacement (or total distance travelled in a specified direction) will be 250 km – 250 km = 0 km. Now,

Average Velocity = \(\frac{\text { Total displacement }}{\text { Total time taken }}\)

= \(\frac{250 \mathrm{~km}-250 \mathrm{~km}}{5 \mathrm{~h}+5 \mathrm{~h}}\)

= \(\frac{0 \mathrm{~km}}{10 \mathrm{~h}}\)

= 0 km/h

Thus, the average velocity of the bus for the whole journey (both ways) is 0 kilometres per hour. No direction can be stated in this case of zero velocity.