Dimensional Formula of Speed

Rest and motion are the two main attributes considered when defining a body’s physical state. As the name suggests, rest means a state of the body when it is static, or no displacement occurs. On the other hand, motion is considered the state where a body moves from one place to another.

Several physical attributes need to be considered to define these states. Of these attributes, dimensional speed is one of the main units without which defining rest and motion or a property’s behaviour in these states become next to impossible. Furthermore, with the study of speed, we can understand its contribution to Newtonian mechanics.

What is Speed and its SI Unit?

Speed is defined as the distance covered by a body from one point to the other in one second. For example, if a body has travelled a distance of ‘d’ and the total time has taken is ‘t’, then the mathematical expression of dimensional speed can be written as: 

s = d/t 

SI and CGS Units of Speed

Dimensional speed depends on distance and time. Therefore, the units involved are metres and seconds in the SI system. So, considering the formula, we can say that:

s = d/t

Or, s = metre/seconds

Or, s = ms-1

In the CGS system, the unit of distance is in centimetres while that of time remains the same, which is seconds. Therefore, in this metric system, dimensional speed has a unit of cm-1. 

Speed—A Scalar Quantity

In physics, there are two types of quantities—scalar and vector. A scalar quantity only has a magnitude, but a vector quantity has both a magnitude and a direction. Speed is considered a scalar unit because when we have to denote it, we only say that a body’s dimensional speed is about x ms-1. We do not say that it travels towards the west or from point A to point B. This is why speed is not subjected to any differential equations or concepts.

Properties of Scalar Units:

• Speed is defined only with a magnitude or a numeric value. 
• It is always a positive quantity.
• Speed can be zero only when the body is at rest. 
• Despite the initial and final positions of the body being the same or different, its speed will never be zero. 

Types of Speed

1. Translational Speed is exhibited by a body while travelling along a straight line. Here, the length between two points is the total distance travelled.
2. Rotational Speed is exhibited by a body moving along a circular path. As the body moves along the circumference, the distance it covers equals the circumference.
3. Instantaneous Speed is the speed of a body at a particular time. For example, consider that the total time taken is divided into four instances—5th second, 10th second, 15th second, and 20th second. The distance covered in the 5th second is considered the body’s instantaneous speed.
4. Average speed is the total distance covered by a body within a given period. For example, consider a body that has travelled 10 metres in 5 seconds, 15 metres in 3 seconds, and 20 metres in 4 seconds. Therefore, the total distance travelled is 45 metres, and the total time taken is 12 seconds. According to the formula, the average speed is (45/12) ms-1.

Derivation and Explanation of the Dimensional Formula of Speed

Speed is expressed as: 

s = d/t

Here, ‘d’ represents distance or length, and ‘t’ represents time. Therefore, ‘d’ is represented by [L] while time is represented by [T]. 

Putting these values in the speed expression gives: 

s = [L]/[T]

s = [L1T-1]

Since no mass is involved in the expression of speed, its dimensional formula can be given as: 

s = [M0L1T-1]

Conclusion

Speed is one of the most crucial attributes that you must consider when dealing with rest and motion. Even though acceleration, force, and other attributes depend on velocity, both speed and velocity have the same magnitude. Apart from this, speed is used in correlation with velocity in the equations of motions required for calculating time, distance, acceleration, and other units based on complex scenarios of real-body movements. Besides, the dimensional speed formula helps study the graph of velocity-time and displacement-time plots.