A softball pitcher throws a softball to a catcher behind home plate. The softball is 3 feet above the ground when it leaves the pitcher's hand at a velocity of 50 feet per second. If the softball's acceleration is [tex]$-16 \, \text{ft/s}^2$[/tex], which quadratic equation models the situation correctly?

A. [tex]$h(t)=50 t^2 - 16 t + 3$[/tex]

B. [tex][tex]$h(t) = -16 t^2 + 50 t + 3$[/tex][/tex]

C. [tex]$3 = -16 t^2 + 50 t + h_0$[/tex]

D. [tex]$3 = 50 t^2 - 16 t + h_0$[/tex]



Answer :

To model the situation of the softball being pitched, we need to form a quadratic equation that describes its height [tex]\( h(t) \)[/tex] at any time [tex]\( t \)[/tex]. The general form of the quadratic equation for the height in projectile motion is given by:

[tex]\[ h(t) = a t^2 + v t + h_0 \][/tex]

Here, [tex]\( a \)[/tex] represents the acceleration due to gravity, [tex]\( v \)[/tex] represents the initial velocity, and [tex]\( h_0 \)[/tex] represents the initial height.

Given the information:
- The initial velocity [tex]\( v \)[/tex] is 50 feet per second.
- The acceleration due to gravity [tex]\( a \)[/tex] is [tex]\( -16 \)[/tex] feet per second squared.
- The initial height [tex]\( h_0 \)[/tex] is 3 feet.

Let's translate these values into the quadratic equation format:

1. The acceleration [tex]\( a \)[/tex] is [tex]\( -16 \)[/tex], which will be multiplied by [tex]\( t^2 \)[/tex].
2. The initial velocity [tex]\( v \)[/tex] is [tex]\( 50 \)[/tex], which will be multiplied by [tex]\( t \)[/tex].
3. The initial height [tex]\( h_0 \)[/tex] is [tex]\( 3 \)[/tex], which will be added as a constant term.

So, plugging the given values into the equation [tex]\( h(t) = a t^2 + v t + h_0 \)[/tex], we get:

[tex]\[ h(t) = -16 t^2 + 50 t + 3 \][/tex]

Now let's examine the different options given in the problem:

1. [tex]\( h(t) = 50 t^2 - 16 t + 3 \)[/tex]
2. [tex]\( h(t) = -16 t^2 + 50 t + 3 \)[/tex]
3. [tex]\( 3 = -16 t^2 + 50 t + h_0 \)[/tex]
4. [tex]\( 3 = 50 t^2 - 16 t + h_0 \)[/tex]

Analyzing these:

- The first option is incorrect because it has the coefficients [tex]\( 50 \)[/tex] and [tex]\( -16 \)[/tex] in the wrong places.
- The third and fourth options are incorrect because they incorrectly place the initial height [tex]\( 3 \)[/tex] on the left side of the equation instead of incorporating it consistently on the right side as the constant term.

Therefore, the correct quadratic equation that models the situation is:

[tex]\[ h(t) = -16 t^2 + 50 t + 3 \][/tex]

This corresponds to option 2.