In the given problem, the student is asked to rewrite the measurements of position, velocity, and acceleration in the standard form $x_{\text{best}} \pm \Delta x$, where $x_{\text{best}}$ is the best estimate and $\Delta x$ is the uncertainty. Let’s break down the values from the table into this form:

Table 2.5:

Measurements of position, velocity, and acceleration:

1. Position, $x$

   • Best estimate: $53.3 \, \text{cm}$
   • Probable range: $53.1 \, \text{cm}$ to $53.5 \, \text{cm}$
   • Uncertainty, $\Delta x = \frac{53.5 - 53.1}{2} = 0.2 \, \text{cm}$
   • Standard form: $x = 53.3 \pm 0.2 \, \text{cm}$

2. Velocity, $v$

   • Best estimate: $-13.5 \, \text{cm/s}$
   • Probable range: $-14.0 \, \text{cm/s}$ to $-13.0 \, \text{cm/s}$
   • Uncertainty, $\Delta v = \frac{14.0 - 13.0}{2} = 0.5 \, \text{cm/s}$
   • Standard form: $v = -13.5 \pm 0.5 \, \text{cm/s}$

3. Acceleration, $a$

   • Best estimate: $93 \, \text{cm/s}^2$
   • Probable range: $90 \, \text{cm/s}^2$ to $96 \, \text{cm/s}^2$
   • Uncertainty, $\Delta a = \frac{96 - 90}{2} = 3 \, \text{cm/s}^2$
   • Standard form: $a = 93 \pm 3 \, \text{cm/s}^2$

Final Results in Standard Form:

1. $x = 53.3 \pm 0.2 \, \text{cm}$
2. $v = -13.5 \pm 0.5 \, \text{cm/s}$
3. $a = 93 \pm 3 \, \text{cm/s}^2$

Would you like more details on the sources of error or an explanation of how uncertainties are calculated?

Here are 5 related questions:

1. How do we determine the uncertainty in a measurement?
2. What is the significance of expressing results in the form $x_{\text{best}} \pm \Delta x$?
3. How can experimental errors affect the accuracy of the measurements?
4. What types of errors commonly occur in physics experiments like this one?
5. How can we minimize uncertainty in such experiments?

Tip: Always ensure that the uncertainty range reflects possible systematic or random errors in the experiment.