To model the height of the rocket after [tex]\( t \)[/tex] seconds using the given parameters, let's start by understanding the provided variables and the general equation of projectile motion.
The general equation of projectile motion is:
[tex]\[ s(t) = g \cdot t^2 + v_0 \cdot t + s_0 \][/tex]
Here:
- [tex]\( g \)[/tex] is the acceleration due to gravity, which is given as [tex]\( -4.9 \, \text{m/s}^2 \)[/tex].
- [tex]\( v_0 \)[/tex] is the initial velocity, which is given as [tex]\( 39.2 \, \text{m/s} \)[/tex].
- [tex]\( s_0 \)[/tex] is the initial height, which is [tex]\( 0 \, \text{m} \)[/tex] since the rocket is launched from the ground.
Substituting these values into the general equation, we get:
[tex]\[ s(t) = -4.9 \cdot t^2 + 39.2 \cdot t + 0 \][/tex]
Because the initial height [tex]\( s_0 \)[/tex] is zero, the equation simplifies to:
[tex]\[ s(t) = -4.9 \cdot t^2 + 39.2 \cdot t \][/tex]
Therefore, the equation that can be used to model the height of the rocket after [tex]\( t \)[/tex] seconds is:
[tex]\[ s(t) = -4.9 t^2 + 39.2 t \][/tex]
Among the given options, this corresponds to the second option:
[tex]\[ s(t) = -4.9 t^2 + 39.2 t \][/tex]
