This is another physics-based problem from Colin Hughes’s *Maths Challenge* website (mathschallenge.net) that may take a bit more thought.

“A firework rocket is fired vertically upwards with a constant acceleration of 4 m/s^{2} until the chemical fuel expires. Its ascent is then slowed by gravity until it reaches a maximum height of 138 metres.

Assuming no air resistance and taking *g* = 9.8 m/s^{2}, how long does it take to reach its maximum height?”

I can never remember the formulas relating acceleration, velocity, and distance, so I always derive them via integration.

See the Fireworks Rocket