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Physics · Ch 1 — Units and Measurement

Checking the Dimensional Consistency of Equations

1.6.1

Checking the Dimensional Consistency of Equations

Checking the Dimensional Consistency of Equations

When you write an equation, every term that is added or subtracted must have the same dimensions. Consider the equation for the distance xx travelled by a particle starting from position x0x_0 with initial velocity v0v_0 at time t=0t = 0 and moving with uniform acceleration aa:

x=x0+v0t+12at2x = x_0 + v_0 t + \frac{1}{2} a t^2

Let us check the dimensions of each term:

  • [x]=[L][x] = [L]
  • [x0]=[L][x_0] = [L]
  • [v0t]=[LT−1][T]=[L][v_0 t] = [L T^{-1}][T] = [L]
  • [12at2]=[LT−2][T2]=[L]\left[\frac{1}{2} a t^2\right] = [L T^{-2}][T^2] = [L]

Every term on the right-hand side has the dimension of length, which matches the left-hand side. The equation is dimensionally correct.

Note

The factor 12\frac{1}{2} is a pure number and has no dimensions. It does not affect the dimensional check at all. This is why dimensional analysis can never determine dimensionless constants — they simply vanish from the dimensional equations. …

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