Question

a rocket launched from rest, steadily increases speed to 41.0m/s over a 4.30s interval before exhausting all the fuel. the rocket continues upward but slows steadily to a stop 2.80s after running out of fuel. how far did the rocket rise while burning fuel? what is the rockets maximum height?

Explanation:

Step1: Find distance during fuel - burning phase

Use the kinematic equation $x = v_{avg}t$, where for uniformly - accelerated motion starting from rest ($v_0 = 0$), $v_{avg}=\frac{v_0 + v}{2}$. Here, $v_0 = 0$, $v = 41.0$ m/s and $t = 4.30$ s. So, $v_{avg}=\frac{0 + 41.0}{2}=20.5$ m/s. Then $x_1=v_{avg}t=20.5\times4.30$.
$x_1 = 88.15$ m

Step2: Find distance after fuel runs out

First, find the acceleration after fuel runs out. Using $v = v_0+a_2t$, where $v = 0$, $v_0 = 41.0$ m/s and $t = 2.80$ s. So, $a_2=\frac{v - v_0}{t}=\frac{0 - 41.0}{2.80}\approx - 14.64$ m/s². Then use the kinematic equation $v^{2}-v_{0}^{2}=2a_2x_2$. Since $v = 0$ and $v_0 = 41.0$ m/s, $x_2=\frac{v^{2}-v_{0}^{2}}{2a_2}=\frac{0 - 41.0^{2}}{2\times(- 14.64)}\approx57.4$ m.

Step3: Calculate maximum height

The maximum height $H=x_1 + x_2$. So, $H=88.15+57.4 = 145.55$ m.

Answer:

How far did the rocket rise while burning fuel? 88.15 m
What is the rocket's maximum height? 145.55 m