3. A frustum is thrown at an angle of [tex]35^{\circ}[/tex] below the horizontal. It takes 3 seconds for the object to reach the ground.

a. Complete the following table:
\begin{tabular}{|l|c|c|}
\hline & horizontal & vertical \\
\hline Initial velocity & -1084 & -5.138 \\
\hline acceleration & [tex]$0 \, m/s^2$[/tex] & [tex]$-9.8 \, m/s^2$[/tex] \\
\hline
\end{tabular}

(show work here)

[tex]
\begin{array}{l}
12 \cos (35) \, m/s \\
12 \sin (35) \, m/s
\end{array}
[/tex]



Answer :

To analyze the motion of the book, let's break down the problem into horizontal and vertical components, using the information provided.

1. Horizontal Components:

- Initial velocity ([tex]\(v_{0_x}\)[/tex]):
The initial velocity horizontally can be found using the cosine of the angle. The given speed is 12 m/s at an angle of [tex]\(35^\circ\)[/tex] below the horizontal.
[tex]\[ v_{0_x} = 12 \cos(35^\circ) \][/tex]
From the given solution:
[tex]\[ v_{0_x} \approx 9.83 \, \text{m/s} \][/tex]

- Acceleration ([tex]\(a_x\)[/tex]):
Since there is no horizontal acceleration:
[tex]\[ a_x = 0 \, \text{m/s}^2 \][/tex]

2. Vertical Components:

- Initial velocity ([tex]\(v_{0_y}\)[/tex]):
The initial velocity vertically can be found using the sine of the angle (note that it will be positive as we consider the downward direction as positive):
[tex]\[ v_{0_y} = 12 \sin(35^\circ) \][/tex]
From the given solution:
[tex]\[ v_{0_y} \approx 6.88 \, \text{m/s} \][/tex]

- Acceleration ([tex]\(a_y\)[/tex]):
The vertical acceleration is due to gravity:
[tex]\[ a_y = -9.8 \, \text{m/s}^2 \][/tex]

Now, summarizing the values in the table form as requested:

[tex]\[ \begin{tabular}{|l|c|c|} \hline & horizontal & vertical \\ \hline Initial velocity & 9.83 \, \text{m/s} & 6.88 \, \text{m/s} \\ \hline Acceleration & 0 \, \text{m/s}^2 & -9.8 \, \text{m/s}^2 \\ \hline \end{tabular} \][/tex]

Thus, the completed table is:
[tex]\[ \begin{tabular}{|l|c|c|} \hline & horizontal & vertical \\ \hline Initial velocity & 9.83 & 6.88 \\ \hline Acceleration & 0 & -9.8 \\ \hline \end{tabular} \][/tex]