Question

A rocket is fired vertically up from rest. If it is designed to maintain a constant upward acceleration of 1.5g, calculate the time t required for it to reach an altitude of 30 km and its velocity at that position.

Answer 1

It is not clear if the rocket "designed to maintain a constant upward acceleration" already takes into account the gravity that will contract its acceleration. Let's assume that acceleration 1.5g is already a resulting acceleration.

For the motion with constant acceleration

$\displaystyle s(t) = s_0 + v_0 t + \frac{at^2}{2}$

$s_0 = 0\,m; v_0 = 0 \, m/s, a =1.5g = 1.5 \cdot 9.81 =14.715 \; m/s^2.$

$\displaystyle t = \sqrt{\frac{2s}{a}} = \sqrt{\frac{2 \cdot30 \cdot 10^3}{14.715}} = 63.85 \; s.$

If acceleration 1.5g is only what engines can provide, then resulting acceleration a = 1.5g - g (because of gravity) = 0.5g. In this case,

$\displaystyle t = \sqrt{\frac{2s}{a}} = \sqrt{\frac{2 \cdot30 \cdot 10^3}{0.5 \cdot 9.81}} = 110.6 \; s.$