From the story. In Hickam's account, several of the boys' early rockets barely rose or failed outright before they began to fly. A rocket that doesn't push harder than it weighs never leaves the ground. (More in October Sky.)
The thrust-to-weight ratio compares the push to the pull: $T/(mg)$. Below 1 the rocket sits on the pad. At 1 it just hovers. Above 1 it rises, and the net acceleration is $(T/W - 1)g$. A high ratio matters for another reason too: the rocket leaves its launch rod faster, and a fast rocket is steadied better by its fins.
Key idea
$$\frac{T}{W} = \frac{T}{mg}, \qquad a = \Big(\frac{T}{W} - 1\Big)g, \qquad v_\text{rod} = \sqrt{2aL}$$ for a rod of length $L$, starting from rest (ignoring friction).
Worked example: Off the rod
A $100\ \text{g}$ rocket with $4.9\ \text{N}$ of thrust rides a $1.0\ \text{m}$ rod. How fast does it leave the rod?
- $W = 0.98\ \text{N}$, so $T/W = 5$ and $a = 4g = 39.2\ \text{m/s}^2$.
- $v = \sqrt{2(39.2)(1.0)} \approx 8.9\ \text{m/s}$.
Answer: $\approx 8.9\ \text{m/s}$
Common mistake
Thinking any thrust lifts a rocket. Thrust less than weight is still a real force; it just doesn't win.