1 Million Means: 1 million in numerical is represented as 10,00,000. The Indian equivalent of a million is ten lakh rupees. It is not a...

1 Million Means: 1 Million in Rupees, Lakhs and Crores

September 13, 2023Speed and velocity have distinct meanings, just as distance and displacement do. Speed is a scalar quantity that describes “the rate at which an item moves.” The pace at which an object travels over a certain distance is known as speed. A fast-moving item moves quickly and covers a considerable distance in a short period of time. A slow-moving item with a low speed, on the other hand, travels a comparatively little amount of distance in the same length of time. A zero speed object is one that does not move at all.

Velocity is a vector quantity that denotes “the rate at which an item changes location.” Consider a person who takes one step forward and one step back at a quick pace, constantly returning to the same starting place. While this would cause a flurry of activity, it would also result in zero velocity. On this page, let us discuss everything about Speed and Velocity. Read further to find more.

Let us first begin with the common aspects in both speed and velocity. In layman’s terms, we define speed or velocity as the *amount of distance covered in a particular time*.

If a car moving at a higher speed means it is moving fast. We say a slow vehicle has less speed. If some object is fast, it means that either-

- i. it covers more distance in some fixed time, or
- ii. it covers the same distance in less time.

Let’s understand from this given example.

** Example:-**The red car takes \(1\;{\rm{h}}\) to travel \(60\;{\rm{km}}.\) The green car also travels \(1\;{\rm{h}}\) but covers only \(40\;{\rm{km}}.\) So, the green car is slower than the red car.

The words “fast”, “slow”, “at rest” are relative. It means that we use the words fast and slow when comparing them with the state of some other objects. For example, a car may be slower than a train but faster than a bicycle.

In daily life, we usually do not bother whether it is speed or velocity. We say the car speed is \(80\;{\rm{km\;}}{{\rm{h}}^{{\rm{ – 1}}}},\) vehicles stuck in a traffic jam were crawling at \(10\;{\rm{km\;}}{{\rm{h}}^{{\rm{ – 1}}}},\) and so on.

Before learning about the exact meaning of speed and velocity, let us learn about the difference between distance and displacement. We define **distance** as the path length covered while traveling from one point to another. For example:-From the given below image, distance is the sum of individual paths along the road. Its value will be given by, \(s = AP + PQ + QR + RS + ST + TB.\) It does not have direction therefore distance is a scalar quantity.

We define **displacement** as the shortest distance between two points. Displacement is also a distance but in an exact straight line. From the below image, displacement is the straight-line distance from starting point \(\left( A \right)\) to ending point \(\left( B \right).\) Displacement is measured in a particular direction, therefore displacement is a vector quantity.

We can define **speed \(\left( v \right)\)** as *the distance traveled per unit time*. It is a scalar quantity.

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

\(s\) is the distance traveled, and \(t\) is the time taken.

**Velocity \(\left( v \right)\)**. We can define velocity as the displacement covered per unit time in a particular direction. It is a vector quantity.

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

\(d\) is the displacement, and

\(t\) is the time taken.

Both speed and velocity have the same unit as metres per second.

\({\rm{Velocity}}\,{\rm{ = }}\frac{{{\rm{displacement}}}}{{{\rm{time}}}}{\rm{m\;}}{{\rm{s}}^{{\rm{ – 1}}}}\)

Speed | Velocity |

Speed is the rate of change of position with time. | Velocity is the rate of change of position with time in a particular direction. |

Speed is a scalar quantity. | Velocity is a vector quantity. |

Speed can only be positive. | Velocity can be positive, zero, or negative. |

Speed can be equal to or more than velocity. | Velocity can be different with the same speed. |

Speed is the distance covered in time. | Velocity is the displacement covered in time. |

A body in motion does not always travel the same distance in the same period. For example, a bus moves slow or fast according to the traffic on the road, and there may be many stops also.

Average speed is the ratio of the total distance to the total time taken for the travel.

\({\rm{average\;speed = }}\frac{{{\rm{total\;distance}}}}{{{\rm{total\;time}}}}\)

Considering the above image,

\({\rm{average\;speed}} = \frac{{AP + PQ + QR + RS + ST + TB}}{{{\rm{total\;time}}}}\)

Average velocity is the ratio of the displacement to the total time taken.

\({\rm{average\;velocity = }}\frac{{{\rm{displacement}}}}{{{\rm{total\;time}}}}\)

Considering the above image,

\({\rm{average\;velocity}} = \frac{{AB}}{{{\rm{total\;time}}}}\)

Instantaneous speed is the speed at that instant of time. For example, the reading of the speedometer of a running car shows its instantaneous speed.

Yes. One of the differences between speed and velocity is that velocity can be zero or negative. It is because velocity is a vector and depends on the direction of movement.

Negative velocity is a *velocity in the opposite direction* to the direction of movement that is considered positive.

Let us understand this with an example:

A person goes for a walk in a rectangular park of length \(100\,{\rm{m}}\) and breadth \(50\,{\rm{m}}.\) From a corner of the park, he walks along the length \(AB\) in \(200\,{\rm{s}}.\) He covers breadth \(BC\) in \(100\,{\rm{s}},\) length \(CD\) in \(250\,{\rm{m}},\) and \(DA\) takes \(150\,{\rm{s}}.\) After one complete walk, he is at his starting position.

\({\rm{average\;speed}} = \frac{{100 + 50 + 100 + 50\;}}{{200 + 100 + 250 + 150\;}} = \frac{{300\;{\rm{m}}}}{{700\;{\rm{s}}}} = 0.43\;{\rm{m\;}}{{\rm{s}}^{ – 1}}\)

The person has come back to his original position by walking the entire perimeter of the rectangle. The displacement is zero.

\({\rm{average\;velocity = }}\frac{{{\rm{displacement}}}}{{{\rm{total\;time}}}} = \frac{0}{{700}} = 0\;{\rm{m\;}}{{\rm{s}}^{ – 1}}\)

Average velocity can be calculated in another way, considering direction.

Let the right and upward direction be considered positive. So, \(A\) to \(B,\) \(B\) to \(C\) is positive displacement (red arrows). This makes the left direction of \(C\) to \(D\) and the downward path of \(D\) to \(A\) negative displacement (blue arrows).

\({\rm{average\;velocity}} = \frac{{100 + 50 – 100 – 50}}{{200 + 100 + 250 + 150}} = \frac{{0\;{\rm{m}}}}{{700\;{\rm{s}}}} = 0\;{\rm{m\;}}{{\rm{s}}^{ – 1}}\)

Suppose there is another pathway extending beyond \(DA,\) and the person walks another \(10\,{\rm{m}}.\) Therefore, the average velocity becomes negative.

\({\rm{average\;velocity}} = \frac{{100 + 50 – 100 – 50 – 10}}{{200 + 100 + 250 + 150}} = \frac{{ – 10\;{\rm{m}}}}{{700\;{\rm{s}}}} = – 0.0142\;{\rm{m\;}}{{\rm{s}}^{ – 1}}\)

We have seen that speed and velocity cannot always be constant. Instead, they change depending on the path a body is moving and the forces that act on the moving body. This takes us to another critical parameter in motion, that is, acceleration.

Acceleration is defined as the *rate of change of velocity*. In other words, we can say that an object accelerates if its speed, its direction, or both changes. A cyclist pushes the pedals of the cycle to move faster. This causes a force to push the cycle with a higher velocity.

Let the velocity, in the beginning, is \(‘u’.\) It increased to velocity, \(‘v’,\) after \(‘t’\) seconds.

Therefore,

\({\rm{acceleration}}\left( a \right) = \;\frac{{{\rm{final\;velocity}}\left( v \right) – {\rm{initial\;velocity}}\left( u \right)}}{{{\rm{time}}\left( t \right)}}\)

Acceleration is a vector and can be positive, zero, or negative.

Unit of acceleration is,

\({\rm{acceleration}} = \frac{{{\rm{change\;in\;velocity}}\;\left( {{\rm{m\;}}{{\rm{s}}^{{\rm{ – 1}}}}} \right)}}{{{\rm{time}}\;\left( s \right)}}{\rm{m\;}}{{\rm{s}}^{ – 2}}\)

- Speed and velocity differ by the inclusion of direction in velocity.
- Speed is distance divided by time, while velocity is displacement divided by time.
- Speed can only be positive. Velocity can be positive, zero, or negative in magnitude.
- A change in velocity is called acceleration.
- A speedometer of a vehicle displays its instantaneous speed.

**A satellite revolves around a small planet in a circular orbit of a radius of \(\;100\;{\rm{km}}.\) It takes \(\;10\;{\rm{h}}\)to complete one revolution. What are its speed and velocity?**Circumference of the orbit is the distance traveled, \(d.\)

Solution:

\(d = 2\pi r = 2 \times \pi \times 100 = 628.31\;{\rm{km}}\)

The time to complete one circle is \(10\,{\rm{h}}{\rm{.}}\)

\({\rm{speed}} = \frac{d}{t} = \frac{{628.31}}{{10}} = 62.831\;{\rm{km\;}}{{\rm{h}}^{ – 1}}\)

After one complete circle, the satellite is at its original position. Therefore, displacement is zero making its average velocity zero.**A car travels west \(40\,{\rm{km}}\) in \(12\,{\rm{h}}.\) Then, it turns right toward the north and runs \(30\,{\rm{km}}\) in \(1.5\,{\rm{h}}.\) What are its average speed and velocity?**

Solution:

\({\rm{average\;speed}} = \frac{{BC + BA}}{{2 + 1.5}} = \frac{{40 + 30}}{{2 + 1.5}} = \frac{{70}}{{3.5}} = 20\;{\rm{km\;}}{{\rm{h}}^{ – 1}}\)

Velocity considers displacement, which is the shortest distance between starting and ending points.

Using Pythagoras theorem,

\( \Rightarrow B{C^2} + B{A^2} = A{C^2}\)

\( \Rightarrow \;\;{40^2} + {30^2} = A{C^2}\)

\( \Rightarrow \;\;AC = 50\)

Hence,

\({\rm{Displacement}} = AC = 50\;{\rm{km}}\)

\({\rm{average\;velocity = }}\frac{{{\rm{displacement}}}}{{{\rm{time}}}} = \frac{{AC}}{t} = \;\frac{{50}}{{3.5}} = 14.28\;{\rm{km\;}}{{\rm{h}}^{ – 1}}\)

Since displacement can never be more than distance, velocity can never be more than speed in magnitude.

** Q.1. When do velocity and speed be equal in numerical value?** If an object moves in a straight line, then displacement and distance are the same. So the numerical value of speed and velocity is the same.

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** Q.2. Are speed and velocity the same?** No, speed is the distance traveled per unit time while velocity is the displacement per unit time. Speed is a scalar, and velocity is a vector, having the direction component.

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** Q.3. How can velocity be negative?** If we take a direction as positive, then its opposite direction is negative. If the distance covered in the negative direction is more, then the average velocity can be negative.

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** Q.4. What does the speedometer of a car show, average or instantaneous speeds?** The speedometer shows the speed at that instant of time that is the instantaneous speeds.

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** Q.5. Why do we say that velocity cannot be more than speed?**Because displacement can be equal to or less than distance. So, velocity can be equal or less, but it can never be more than speed.

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