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Section 7.6 Projectile Motion
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?
Figure 7.6.1. Three marbles launched off a table.
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*}
