What is the Max height of a rocket?

Question

A rocket accelerates vertically with a constant acceleration of 21.1m/s/s until it’s engine fails after 50 seconds. The rocket reaches an altitude of 26375m when the engine fails. When does the rocket reach its max height? (Ignoring air resistance)

Answer 1

First let's check some numbers.

One useful formula is: d = 1/2 * a * t²

This is most often applied to free-fall, but also applies to anything starting from 0 velocity and travelling with constant acceleration.

From your numbers:

d = 1/2 * 21.1 * 50² = 26375 (So far, so good!!)

Next, recognize that when the rocket fails, it has velocity and will continue to climb until the velocity reaches 0. Since we are just asked for the time, use the formula v = v_i - g*t (g is the acceleration of gravity: 9.8 m/s²)

First we need v_i (The velocity when the engine fails). This is simply a*t, or 21.1 * 50 = 1055 m/s

So: v = v_i - g*t = 1055 - 9.8t

We want the time to maximum height, at which time v = 0

So, 0 = 1055 - 9.8t, and t = 1055/9.8 = 107.65 seconds (Add 50 to get the total time since launch)