2.7 Falling objects  (Page 5/9)

Suppose the ball falls 1.0000 m in 0.45173 s. Assuming the ball is not affected by air resistance, what is the precise acceleration due to gravity at this location?

Strategy

Draw a sketch.

We need to solve for acceleration $a$ . Note that in this case, displacement is downward and therefore negative, as is acceleration.

Solution

1. Identify the knowns. $y_0=0$ ; $y=\mathrm{–1}\text{.0000 m}$ ; $t=0\text{.45173}$ ; $v_0=0$ .

2. Choose the equation that allows you to solve for $a$ using the known values.

3. Substitute 0 for $v_0$ and rearrange the equation to solve for $a$ . Substituting 0 for $v_0$ yields

Solving for $a$ gives

4. Substitute known values yields

so, because $a=-g$ with the directions we have chosen,

Discussion

The negative value for $a$ indicates that the gravitational acceleration is downward, as expected. We expect the value to be somewhere around the average value of $9\text{.}{\text{80 m/s}}^2$ , so $9\text{.}{\text{8010 m/s}}^2$ makes sense. Since the data going into the calculation are relatively precise, this value for $g$ is more precise than the average value of $9\text{.}{\text{80 m/s}}^2$ ; it represents the local value for the acceleration due to gravity.

Phet explorations: equation grapher

Learn about graphing polynomials. The shape of the curve changes as the constants are adjusted. View the curves for the individual terms (e.g. $y=\text{bx}$ ) to see how they add to generate the polynomial curve.

Section summary

• An object in free-fall experiences constant acceleration if air resistance is negligible.
• On Earth, all free-falling objects have an acceleration due to gravity $g$ , which averages
  $g=9\text{.}{\text{80 m/s}}^2.$
• Whether the acceleration a should be taken as $+g$ or $-g$ is determined by your choice of coordinate system. If you choose the upward direction as positive, $a=-g=-9\text{.}\text{80 m}{\text{/s}}^2$ is negative. In the opposite case, $a=\mathrm{+g}=9\text{.}{\text{80 m/s}}^2$ is positive. Since acceleration is constant, the kinematic equations above can be applied with the appropriate $+g$ or $-g$ substituted for $a$ .
• For objects in free-fall, up is normally taken as positive for displacement, velocity, and acceleration.

Conceptual questions