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A coach throws a football from a height of 4 feet with an initial velocity of 59 feet per second at an elevation angle of [tex]$45^{\circ}$[/tex]. Which parametric equations represent the path of the football?

A. [tex]$x(t)=59 \sin \left(45^{\circ}\right) t$[/tex] and [tex]$y(t)=-16 t^2+59 \cos \left(45^{\circ}\right) t$[/tex]

B. [tex]$x(t)=59 \cos \left(45^{\circ}\right) t$[/tex] and [tex]$y(t)=-16 t^2+59 \sin \left(45^{\circ}\right) t$[/tex]

C. [tex]$x(t)=59 \cos \left(45^{\circ}\right) t$[/tex] and [tex]$y(t)=-16 t^2+59 \sin \left(45^{\circ}\right) t+4$[/tex]

D. [tex]$x(t)=59 \sin \left(45^{\circ}\right) t$[/tex] and [tex]$y(t)=-16 t^2+59 \cos \left(45^{\circ}\right) t+4$[/tex]

Answer :

To solve the problem of finding the parametric equations for the path of the football, let's break it down step-by-step:

1. Understand the Components:
- The football is thrown from a height of 4 feet.
- The initial velocity is 59 feet per second.
- The angle of elevation is [tex]\(45^\circ\)[/tex].

2. Parametric Equations Basics:
- Parametric equations for projectile motion are given by:
- [tex]\( x(t) = v_0 \cos(\theta) \cdot t \)[/tex]
- [tex]\( y(t) = -\frac{1}{2} g \cdot t^2 + v_0 \sin(\theta) \cdot t + h_0 \)[/tex]
- Here, [tex]\(v_0\)[/tex] is the initial velocity, [tex]\(\theta\)[/tex] is the angle of projection, [tex]\(g\)[/tex] is the acceleration due to gravity (32 feet per second[tex]\(^2\)[/tex] for Earth's gravity), and [tex]\(h_0\)[/tex] is the initial height.

3. Calculate Components:
- Convert angles from degrees to radians if needed, but [tex]\(\sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}\)[/tex].

4. Horizontal Component (x(t)):
- The horizontal component does not involve gravity.
- [tex]\( x(t) = v_0 \cos(45^\circ) \cdot t = 59 \cdot \frac{\sqrt{2}}{2} \cdot t \)[/tex]
- Which simplifies to: [tex]\( x(t) = 59 \cos(45^\circ) \cdot t \)[/tex]

5. Vertical Component (y(t)):
- The vertical equation considers gravity and initial height.
- [tex]\( y(t) = -16 t^2 + 59 \sin(45^\circ) \cdot t + 4 \)[/tex]
- Here, [tex]\(-16\)[/tex] comes from [tex]\(-\frac{1}{2} \cdot 32\)[/tex], the gravitational effect.

6. Combine Results:
- Which gives us the vertical component: [tex]\( y(t) = -16 t^2 + 59 \sin(45^\circ) \cdot t + 4 \)[/tex]

Now, let's compare these equations with the given options. The correct option should match this setup:

- Correct Choice:
- [tex]\( x(t) = 59 \cos(45^\circ) t \)[/tex]
- [tex]\( y(t) = -16 t^2 + 59 \sin(45^\circ) t + 4 \)[/tex]

So, the correct parametric equations are:
- [tex]\( x(t) = 59 \cos(45^\circ) t \)[/tex]
- [tex]\( y(t) = -16 t^2 + 59 \sin(45^\circ) t + 4 \)[/tex]

Therefore, the answer is the third option:
[tex]\(x(t)=59 \cos \left(45^{\circ}\right) t\)[/tex] and [tex]\(y(t)=-16 t^2+59 \sin \left(45^{\circ}\right) t+4\)[/tex]

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Rewritten by : Barada