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The equation for a projectile's height versus time is [tex]h(t)=-16 t^2+V_0 t+h_0[/tex].

A tennis ball machine serves a ball vertically into the air from a height of 2 feet, with an initial speed of 130 feet per second. Which equation correctly models the ball's height as a function of time?

A. [tex]h(t)=-16 t^2+2 t+130[/tex]
B. [tex]h(t)=-16 t^2-130 t+2[/tex]
C. [tex]h(t)=-16 t^2+130 t+2[/tex]
D. [tex]h(t)=-16 t^2-2 t+130[/tex]



Answer :

GinnyAnswer
Let's solve the problem step-by-step.

We are given:
- The initial height [tex]\( h_0 = 2 \)[/tex] feet.
- The initial velocity [tex]\( V_0 = 130 \)[/tex] feet per second.

The general equation for the projectile's height [tex]\( h(t) \)[/tex] as a function of time [tex]\( t \)[/tex] is given by:
[tex]\[ h(t) = -16t^2 + V_0 t + h_0 \][/tex]

We need to substitute the given values of [tex]\( V_0 \)[/tex] and [tex]\( h_0 \)[/tex] into this equation.

Step 1: Substitute [tex]\( V_0 = 130 \)[/tex] into the equation.
[tex]\[ h(t) = -16t^2 + 130t + h_0 \][/tex]

Step 2: Substitute [tex]\( h_0 = 2 \)[/tex] into the equation.
[tex]\[ h(t) = -16t^2 + 130t + 2 \][/tex]

So the equation that models the ball's height as a function of time is:
[tex]\[ h(t) = -16t^2 + 130t + 2 \][/tex]

Now, let's match this with the given options:

A. [tex]\( h(t) = -16t^2 + 2t + 130 \)[/tex]
- This equation incorrectly places the initial height and the initial velocity coefficients in the wrong order.

B. [tex]\( h(t) = -16t^2 - 130t + 2 \)[/tex]
- This equation incorrectly has a negative initial velocity term [tex]\( -130t \)[/tex].

C. [tex]\( h(t) = -16t^2 + 130t + 2 \)[/tex]
- This equation is correct and matches our derived equation.

D. [tex]\( h(t) = -16t^2 - 2t + 130 \)[/tex]
- This equation incorrectly places the initial height and the initial velocity coefficients in the wrong order and signs.

Therefore, the correct equation is:
[tex]\[ c. h(t) = -16t^2 + 130t + 2 \][/tex]

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