From the story. The boys launched rocket after rocket from Cape Coalwood, their name for an old slack dump outside town. A graph turns the numbers from a flight into a picture you can read at a glance. (More in October Sky.)
A velocity-time graph tells the whole story of a straight-up flight. While the motor burns, the line climbs. After burnout, gravity alone acts, so the line falls at a steady slope of $-9.8\ \text{m/s}^2$. The rocket is still going up until that line reaches zero.
Key idea
On a velocity-time graph: slope = acceleration, area between the line and the time axis = displacement (height gained), and apogee is where $v = 0$. For a straight-line boost from rest to $v_b$ at $t_b$, then a coast, the whole area to apogee is one triangle: $h = \tfrac12 v_b\,t_\text{top}$.
Worked example: Read the graph
Velocity rises in a straight line to $60\ \text{m/s}$ at $t = 2\ \text{s}$, then coasts. Find the time of apogee and the height reached.
- After burnout, $v$ falls $9.8\ \text{m/s}$ each second: $60/9.8 = 6.12\ \text{s}$ more, so apogee is at $t = 8.12\ \text{s}$.
- Area of the triangle: $\tfrac12 (8.12)(60) = 244\ \text{m}$.
Answer: $t \approx 8.1\ \text{s}$, $h \approx 244\ \text{m}$
Common mistake
Thinking the rocket is highest at burnout because that's where the velocity graph peaks. The velocity is highest there; the height keeps growing until velocity reaches zero.