1.7 Falling objects  (Page 3/10)

Discussion

The rock is 8.10 m above its starting point at $t=1\text{.}\text{00}$ s, since $y_1>y_0$ . It could be moving up or down; the only way to tell is to calculate $v_1$ and find out if it is positive or negative.

Solution for Velocity $v_1$

1. Identify the knowns. We know that $y_0=0$ ; $v_0=\text{13}\text{.}\text{0 m/s}$ ; $a=-g=-9\text{.}{\text{80 m/s}}^2$ ; and $t=1\text{.}\text{00 s}$ . We also know from the solution above that $y_1=8\text{.}\text{10 m}$ .

2. Identify the best equation to use. The most straightforward is $v=v_0-\text{gt}$ (from $v=v_0+\text{at}$ , where $a=\text{gravitational acceleration}=-g$ ).

3. Plug in the knowns and solve.

Discussion

The positive value for $v_1$ means that the rock is still heading upward at $t=1\text{.}\text{00}\text{s}$ . However, it has slowed from its original 13.0 m/s, as expected.

Solution for Remaining Times

The procedures for calculating the position and velocity at $t=2\text{.}\text{00}\text{s}$ and $3\text{.}\text{00 s}$ are the same as those above. The results are summarized in [link] and illustrated in [link] .

Graphing the data helps us understand it more clearly.

Discussion

The interpretation of these results is important. At 1.00 s the rock is above its starting point and heading upward, since $y_1$ and $v_1$ are both positive. At 2.00 s, the rock is still above its starting point, but the negative velocity means it is moving downward. At 3.00 s, both $y_3$ and $v_3$ are negative, meaning the rock is below its starting point and continuing to move downward. Notice that when the rock is at its highest point (at 1.5 s), its velocity is zero, but its acceleration is still $-9\text{.}{\text{80 m/s}}^2$ . Its acceleration is $-9\text{.}{\text{80 m/s}}^2$ for the whole trip—while it is moving up and while it is moving down. Note that the values for $y$ are the positions (or displacements) of the rock, not the total distances traveled. Finally, note that free-fall applies to upward motion as well as downward. Both have the same acceleration—the acceleration due to gravity, which remains constant the entire time. Astronauts training in the famous Vomit Comet, for example, experience free-fall while arcing up as well as down, as we will discuss in more detail later.