State and explain two dimensional analysis in physics

Example 1

Check the consistency of the equation

x = x0 + v0t + (1/2) at2

where x and x0 are distances, t is time, v is velocity and a is an acceleration of the body.

Now to check if the above equation is dimensionally correct, we have to prove that dimensions of physical quantities are same on both sides. Also, we have to keep in mind that quantities can only be added or subtracted if their dimensions are same.

x = distance = [L]

x0= distance = [L]

v0t = velocity × time = [LT-1] × [T] = [L]

at2 = acceleration × time2 = [LT-2] × [T2] = [L]

Since dimensions of left hand side equals to dimension on right hand side, equation is said to be consistent and dimensionally correct.

Example 2

Check whether the given equation is dimensionally correct.

W = 1/2 mv2 – mgh

where W stands for work done, m means mass, g stands for gravity, v for velocity and h for height.

To check the above equation as dimensionally correct, we first write dimensions of all the physical quantities mentioned in the equation.

W = Work done = Force × Displacement = [MLT-2] × [L] = [ML2T-2]

1/2 mv2 = Kinetic Energy = [M] × [L2T-2] = [ML2T-2]

mgh = Potential Energy = [M] × [LT-2] × [L] = [ML2T-2]

Since all the dimensions on left and right sides are equal it is a dimensionally correct equation.