It takes 4.3 seconds for the rocket to return to earth.
The equation is:
[tex]0=-9.8t^2+v_0t+h_0[/tex]
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:
[tex]0=-9.8t^2+42t[/tex]
We will use the quadratic formula to solve this. The quadratic formula is:
[tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
Plugging in our information we have:
[tex]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[/tex]
x=0 is when the rocket is launched; x=4.3 is when the rocket lands.
