How to Calculate Dimensions of Speed with Examples
The dimensions of speed are fundamental in physics, describing how this quantity relates to the basic units of measurement: mass, length, and time. Understanding the dimensional formula of speed enables analysis, verification, and comparison of physical relationships across different systems of units.
Definition of Speed and Its Dimensional Formula
Speed is defined as the rate at which an object covers distance with respect to time. Mathematically, it is expressed as the ratio of distance travelled to the time taken.
The dimensional formula of speed is derived from its definition. Distance is a fundamental quantity with the dimension of length $(\text{L})$, and time has the dimension $(\text{T})$.
Therefore, speed $= \dfrac{\text{Distance}}{\text{Time}}$. The dimensional formula is $[\text{M}^0\text{L}^1\text{T}^{-1}]$, where mass (M) is not involved.
Derivation of Dimensional Formula of Speed
Distance, or displacement, has the dimensional formula $[\text{M}^0\text{L}^1\text{T}^0]$. Time is expressed as $[\text{M}^0\text{L}^0\text{T}^1]$. Using these, the dimensional formula of speed is obtained by dividing the former by the latter.
$\text{Speed} = \text{Distance} \times (\text{Time})^{-1} = [\text{M}^0\text{L}^1\text{T}^0] \times [\text{M}^0\text{L}^0\text{T}^{-1}] = [\text{M}^0\text{L}^1\text{T}^{-1}]$
Comparison of Speed and Velocity Dimensions
Both speed and velocity have identical dimensions as their formulae only differ by direction. The SI unit for both quantities is metre per second $(\mathrm{m \, s^{-1}})$.
The dimension $[\text{M}^0\text{L}^1\text{T}^{-1}]$ applies to any form of speed or velocity irrespective of the specific context.
Dimensional Formula of Speed of Light
The speed of light in a vacuum is a universal constant and is denoted by $c$. Like any form of speed, its dimensional formula is $[\text{M}^0\text{L}^1\text{T}^{-1}]$.
The numerical value of $c$ is approximately $3 \times 10^8 \, \mathrm{m \, s^{-1}}$, but the dimensional formula depends only on fundamental units.
For related concepts, see Dimensions Of Impedance.
Table: Dimensional Formulae of Related Physical Quantities
| Physical Quantity | Dimensional Formula |
|---|---|
| Speed / Velocity | $[\text{M}^0\text{L}^1\text{T}^{-1}]$ |
| Displacement / Distance | $[\text{M}^0\text{L}^1\text{T}^0]$ |
| Time | $[\text{M}^0\text{L}^0\text{T}^1]$ |
| Force | $[\text{M}^1\text{L}^1\text{T}^{-2}]$ |
Applications of Dimensional Analysis in Speed
Dimensional analysis is used to confirm the dimensional consistency of physical equations involving speed. It ensures mathematical equations are valid across any system of units.
The process assists in converting units and comparing the physical quantities in mechanics, such as when analysing kinetic energy or momentum. For reference, see Dimensions Of Momentum.
Examples Using the Dimensional Formula of Speed
Speed can be used to derive and verify equations. For example, if a wave equation expresses speed as $v = f \lambda$, the dimensions on both sides must agree as $[\text{L}^1\text{T}^{-1}]$.
Dimensional analysis can also identify errors in physical formulae and deduce possible forms of unknown relations.
Summary of Key Properties
- Dimensional formula of speed is $[\text{M}^0\text{L}^1\text{T}^{-1}]$
- Speed and velocity share the same dimensions
- Unit analysis ensures equation correctness
- Dimensions enable conversion between unit systems
- Used in verification of physical laws in mechanics
Related Topics in Dimensional Analysis
A clear understanding of dimensions is useful for analysing other physical quantities. Topics such as Dimensions Of Work, Dimensions Of Density, and Dimensions Of Force provide systematic steps related to dimensional formulae for a wide range of quantities.
FAQs on Understanding the Dimensions of Speed in Physics
1. What are the dimensions of speed?
Speed is a physical quantity that expresses how quickly an object moves and its dimensions are given in terms of the fundamental units of length and time.
- The dimensional formula of speed is [M0 L1 T-1].
- It is derived from speed = distance/time (distance in metres, time in seconds).
- This shows speed has no mass (M0), one length (L1), and is inversely proportional to time (T-1).
2. Why is the dimensional formula of speed written as M0L1T-1?
The dimensional formula of speed is written as [M0L1T-1] because speed only depends on length and time.
- No involvement of mass (M0).
- Length appears as numerator (L1).
- Time is in the denominator (T-1).
3. What is the SI unit of speed and its dimensions?
The SI unit of speed is metre per second (m/s) and its dimensions are [M0L1T-1].
- Speed measures how fast something moves in one second.
- It combines distance (metre) and time (second) in its definition.
4. How are the dimensions of velocity different from speed?
Speed and velocity share the same dimensional formula: [M0L1T-1]. The difference lies in direction:
- Speed is a scalar quantity (no direction).
- Velocity is a vector quantity (has direction).
5. What physical quantities have the same dimensions as speed?
Quantities with the same dimensions as speed include
- Velocity (vector, but dimensional formula is the same: [M0L1T-1])
- Wave speed (c)
- Drift velocity of electrons
6. How can dimensional analysis help check the correctness of speed formulas?
You can use dimensional analysis to verify if a formula for speed (or any physical quantity) is correct by comparing the dimensions on both sides.
- First, find dimensions for each term in the formula.
- Check if the resultant dimensions match [M0L1T-1].
- If not, the formula is dimensionally incorrect.
7. Is speed a scalar or vector quantity? What are its dimensions?
Speed is a scalar quantity because it has only magnitude, not direction. Its dimensions are [M0L1T-1].
- It measures how fast an object moves, regardless of the direction.
- Often compared with velocity (which is a vector).
8. What are the fundamental quantities involved in the dimensions of speed?
The fundamental quantities that form the basis of speed’s dimensions are
- Length (L): covers distance travelled
- Time (T): measures duration of motion
9. Can you give examples of how to calculate the dimensional formula of speed?
To calculate the dimensional formula of speed:
- Use the definition, speed = distance/time.
- Distance has the dimension L.
- Time has the dimension T.
- Therefore, the dimensional formula is L1T-1 (or [M0L1T-1]).
10. What is the importance of knowing the dimensions of speed in physics?
Knowing the dimensions of speed helps students and scientists:
- Check whether equations involving speed are dimensionally correct.
- Convert units from one system to another.
- Find relationships between physical quantities in formulas.
- Quickly identify errors or inconsistencies in calculations.
11. Define speed and give its dimensional formula.
Speed is defined as the rate of change of position of an object with respect to time. Its dimensional formula is [M0L1T-1], representing length per unit time.
12. What is the relationship between speed and velocity in terms of dimensions?
Both speed and velocity have the same dimensions [M0L1T-1], as both represent distance covered per unit time; the difference lies in that velocity also includes direction.
