To determine the equation for the height of Elena's rocket, we need to consider a few key aspects of the problem: the initial height, the initial velocity, and how gravity affects the rocket's height over time.
1. Initial Height ([tex]\(h_0\)[/tex]): Both Justin and Elena launched their rockets from the same position, which is 2 feet above the ground. Therefore, the initial height for both rockets is [tex]\( h_0 = 2 \)[/tex] feet.
2. Initial Velocity (v): Justin's rocket has an initial velocity of 60 feet per second. Elena's rocket has an initial velocity that is double that of Justin's. Therefore, the initial velocity for Elena's rocket is:
[tex]\[
v = 2 \times 60 = 120 \text{ feet per second}
\][/tex]
3. Gravity's Effect ([tex]\(a\)[/tex]): Gravity affects both rockets equally and is given by [tex]\(-16t^2\)[/tex] (where [tex]\(t\)[/tex] is the time in seconds).
Putting all this together, we form the equation for the height of Elena's rocket [tex]\(h(t)\)[/tex]:
[tex]\[
h(t) = -16t^2 + 120t + 2
\][/tex]
Among the provided options:
- [tex]\( h = -16 t^2 + 60 t + 4 \)[/tex]
- [tex]\( h = -32 t^2 + 120 t + 4 \)[/tex]
- [tex]\( h = -32 t^2 + 60 t + 2 \)[/tex]
- [tex]\( h = -16 t^2 + 120 t + 2 \)[/tex]
We compare our derived equation [tex]\( h(t) = -16t^2 + 120t + 2 \)[/tex] to the provided options. The correct equation that matches our derived function for the height of Elena's rocket is:
[tex]\[
h = -16 t^2 + 120 t + 2
\][/tex]
