The projectile motion of an object can be modeled using [tex]\( s(t) = g t^2 + v_0 t + s_0 \)[/tex], where:
- [tex]\( g \)[/tex] is the acceleration due to gravity,
- [tex]\( t \)[/tex] is the time in seconds since launch,
- [tex]\( s(t) \)[/tex] is the height after [tex]\( t \)[/tex] seconds,
- [tex]\( v_0 \)[/tex] is the initial velocity,
- [tex]\( s_0 \)[/tex] is the initial height.

The acceleration due to gravity is [tex]\(-4.9 \, m/s^2\)[/tex].

A rocket is launched from the ground at an initial velocity of 39.2 meters per second. Which equation can be used to model the height of the rocket after [tex]\( t \)[/tex] seconds?

A. [tex]\( s(t) = -4.9 t^2 + 39.2 \)[/tex]
B. [tex]\( s(t) = -4.9 t^2 + 39.2 t \)[/tex]
C. [tex]\( s(t) = -4.9 t^2 + 39.2 t + 39.2 \)[/tex]
D. [tex]\( s(t) = -4.9 t^2 + 39.2 t - 39.2 \)[/tex]



Answer :

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]