Dimensions and Dimensional analysis question

[Zerocool97]

In this formula, E = 1/2 mv²

1/2 is not considered as it does not have any dimension. But then does it not affect the formula. Why? Explain! Please

 

[Simon Bridge]

Welcome to PF;
Of course the value of 1/2 affects the formula... what is there to explain?

 

[dauto]

That's a limitation of dimensional analysis. numerical factors such as 1/2 in the formula for the kinetic energy cannot be calculated by dimensional analysis. Some more advanced calculation method must be used.

 

[SW VandeCarr]

In this formula, E = 1/2 mv²

1/2 is not considered as it does not have any dimension. But then does it not affect the formula. Why? Explain! Please

Dimensional analysis is generally done with a few chosen "basic" dimensions. For most problems in Newtonian mechanics, length, mass and time (LMT) are sufficient. For energy, dimensions are: $E=ML^{2}T^{-2}$. The dimensional formula does not distinguish between kinetic and potential energy. Dimensionless values such as 1/2 in the KE formula do not appear in the dimensional formula.

Also, dimensional formulas do not distinguish between energy density and pressure. Both are represented by the formula $ML^{-1}T^{-2}$. The difference is that energy density is a scalar quantity and pressure is a vector quantity.

DA is useful in showing relationships between different formulas used in physics. That doesn't mean that different formulas with the same dimensional formula are necessarily strictly equivalent. They may only be dimensionally equivalent.

 

[pmacias]

note:

In this formula, E = 1/2 mv², the term 1/2 is a constant and does not have any dimension. This means that it does not have any physical unit associated with it. However, it still plays an important role in the formula.

The 1/2 in the formula represents the coefficient of the kinetic energy (E) term. This coefficient is necessary to calculate the exact amount of kinetic energy possessed by an object with a given mass (m) and velocity (v). Without this coefficient, the formula would not accurately represent the relationship between kinetic energy and mass/velocity.

Furthermore, dimensional analysis is a powerful tool used in science to check the validity of equations and to ensure that all terms have the correct units. In this case, the dimensions of the left and right side of the equation match, with both sides representing units of energy. Therefore, the inclusion of the 1/2 term does not affect the dimensional consistency of the formula.

In conclusion, although the 1/2 term does not have any dimension, it is still a crucial part of the formula and is necessary for accurately calculating the kinetic energy of an object. Its inclusion does not affect the validity or consistency of the equation.