Activity 7.6.1. Three Marbles.
You throw three marbles off a table, each with the same initial speed but a different direction. Before you do any calculations, make the following predictions:
- Which is in the air for the most time?
- Which travels the most horizontal distance?
- Which is moving fastest when it hits?
Start with Marble 2.
- If the height of the table is \(h\text{,}\) how much time does it take for marble 2 to hit the ground?
- How far has the marble traveled horizontally in that amount of time?
- What is the marble’s speed when it hits the ground?
Now try Marble 1.
- If the angle is \(\theta\text{,}\) how long does it take for marble 1 to hit the ground?
- How far has the marble traveled horizontally in that amount of time?
- What is the marble’s speed when it hits the ground?
Last of all, make sense of the solution!
Solution.
\begin{equation*}
t = \frac{v}{g}\sin\theta \pm \sqrt{\frac{v^2}{g^2}\sin^2\theta + 2\frac{h}{g}}
\end{equation*}
\begin{equation*}
x = vt\cos\theta
\end{equation*}
\begin{equation*}
v_f = \sqrt{v^2 + 2gh}
\end{equation*}
