The simple equation \$d/v = 1/H_0\$ gives a rough estimate of how long the Universe has been expanding.
Think of it like this: if you know how fast a balloon is inflating (the speed of galaxies moving apart) and how far it has already stretched, you can back‑track to when it started from a tiny point.
\$H_0\$ (Hubble constant) is measured in units of velocity per distance, e.g. km s⁻¹ Mpc⁻¹.
Rearranging \$d/v = 1/H_0\$ gives:
\$\text{Age} \approx \frac{1}{H_0}\$
This is a *first‑order* estimate – it assumes the expansion rate has been constant, which isn’t exactly true, but it’s a great starting point for exams.
Suppose the accepted value is \$H_0 = 70 \text{ km s}^{-1}\text{Mpc}^{-1}\$.
First, convert 1 Mpc to kilometres:
\$1\,\text{Mpc} \approx 3.086 \times 10^{19}\,\text{km}\$
Then:
\$\text{Age} \approx \frac{1}{70}\,\frac{\text{Mpc}}{\text{km s}^{-1}} = \frac{3.086 \times 10^{19}\,\text{km}}{70\,\text{km s}^{-1}}\$
\$\text{Age} \approx 4.4 \times 10^{17}\,\text{s}\$
Convert seconds to years (1 yr ≈ 3.16 × 10⁷ s):
\$\text{Age} \approx \frac{4.4 \times 10^{17}}{3.16 \times 10^{7}} \approx 1.4 \times 10^{10}\,\text{yr}\$
So the Universe is about 14 billion years old – a figure that matches modern observations. 🚀
Remember:
| Hubble Constant (H₀) | Age Estimate (1/H₀) |
|---|---|
| 70 km s⁻¹ Mpc⁻¹ | ≈ 14 billion years |
| 67 km s⁻¹ Mpc⁻¹ | ≈ 15 billion years |
| 73 km s⁻¹ Mpc⁻¹ | ≈ 13 billion years |