Question

the initial vertical velocity of a rocket shot straight up is 42 meters per second. How long does it take for the rocket to return to earth? round your answer to the nearest tenth of a second. show all your work.

Answer 1

It takes 4.3 seconds for the rocket to return to earth.

The equation is:
$0=-9.8t^2+v_0t+h_0$
where -9.8m/sec² is the acceleration due to gravity, v₀ is the initial velocity, and h₀ is the initial height.  We will go from the assumption that the rocket is launched from the ground, so h₀=0, and we are told that the initial velocity, v₀, is 42.  This gives us:

$0=-9.8t^2+42t$

We will use the quadratic formula to solve this.  The quadratic formula is:
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Plugging in our information we have:
$t=\frac{-42\pm \sqrt{42^2-4(-9.8)(0)}}{2(-9.8)} \\ \\=\frac{-42\pm \sqrt{1764-0}}{-19.6} \\ \\=\frac{-42\pm \sqrt{1764}}{-19.6} \\ \\=\frac{-42\pm 42}{-19.6}=\frac{-42-42}{-19.6} \text{ or } \frac{-42+42}{-19.6} \\ \\=\frac{-84}{-19.6}\text{ or }\frac{0}{-19.6} \\ \\=4.3\text{ or }0$

x=0 is when the rocket is launched; x=4.3 is when the rocket lands.