36.8.3 The Binomial Approximation

You’ve met the binomial approximation earlier in this text and in your calculus class. The binomial approximation is useful when we need to calculate a relativistic expression for a nonrelativistic velocity $v ≪ c$ . Because $v^{2} / c^{2} ≪ 1$ in these cases, we can write

If v ≪ c : {\begin{array}{l} \sqrt{1 - β^{2}} = {(1 - \frac{v^{2}}{c^{2}})}^{1 / 2} \approx 1 - \frac{1}{2} \frac{v^{2}}{c^{2}} \\ γ = \frac{1}{\sqrt{1 - β^{2}}} = {(1 - \frac{v^{2}}{c^{2}})}^{- 1 / 2} \approx 1 + \frac{1}{2} \frac{v^{2}}{c^{2}} \end{array}

(36.26)

The following example illustrates the use of the binomial approximation.

The binomial approximation

x ≪ 1

, then

{(1 + x)}^{n} \approx 1 + n x

EXAMPLE 36.8 The shrinking school bus

An 8.0-m-long school bus drives past at $30 m / s$ . By how much is its length contracted?

MODEL The school bus is at rest in an inertial reference frame $S'$ moving at velocity $v = 30 m / s$ relative to the ground frame S. The given length, 8.0 m, is the proper length $ℓ$ in frame $S'$ .

SOLVE In frame S, the school bus is length contracted to

L = \sqrt{1 - β^{2}} ℓ

The bus’s velocity v is much less than c, so we can use the binomial approximation to write

L \approx (1 - \frac{1}{2} \frac{v^{2}}{c^{2}}) ℓ = ℓ - \frac{1}{2} \frac{v^{2}}{c^{2}} ℓ

The amount of the length contraction is

\begin{array}{l} ℓ - L & = \frac{1}{2} \frac{v^{2}}{c^{2}} ℓ = \frac{1}{2} {(\frac{30 m/s}{3.0 \times 10^{8} m/s})}^{2} (8.0 m) \\ = 4.0 \times 10^{- 14} m = 40 fm \end{array}

where $1 fm = 1 f e m t o m e t e r = 10^{- 15} m$ .

REVIEW The bus “shrinks” by only slightly more than the diameter of the nucleus of an atom. It’s no wonder that we’re not aware of length contraction in our everyday lives. If you had tried to calculate this number exactly, your calculator would have shown $ℓ - L = 0$ because the difference between $ℓ$ and L shows up only in the 14th decimal place. A scientific calculator determines numbers to 10 or 12 decimal places, but that isn’t sufficient to show the difference. The binomial approximation provides an invaluable tool for finding the very tiny difference between two numbers that are nearly identical.