To estimate the reaction time of a ruler dropped under free fall, we can use principles from kinematics. Here’s a step-by-step solution, including recent examples for clarity:

1. Understanding Free Fall and Kinematics

In free fall, the distance traveled $dd$d can be described by the equation: $d=\frac{1}{2}g{t}^{2}d\; =\; \backslash frac\{1\}\{2\}\; g\; t^2$d=21​gt2 where $gg$g is the acceleration due to gravity (approximately $9.8\hspace{0.17em}{2}^{}9.8\; \backslash ,\; \backslash text\{m/s\}^2$9.8m/s2) and $tt$t is the time in seconds.

2. Calculating the Time for Free Fall

Given:

• Distance traveled $d=21\hspace{0.17em}cm=0.21\hspace{0.17em}md\; =\; 21\; \backslash ,\; \backslash text\{cm\}\; =\; 0.21\; \backslash ,\; \backslash text\{m\}$d=21cm=0.21m

We need to find $tt$t. Rearranging the formula to solve for $tt$t: $t=\sqrt{\frac{2d}{g}}t\; =\; \backslash sqrt\{\backslash frac\{2d\}\{g\}\}$t=g2d​​ Substitute the values: $t=\sqrt{\frac{2\times 0.21}{9.8}}t\; =\; \backslash sqrt\{\backslash frac\{2\; \backslash times\; 0.21\}\{9.8\}\}$t=9.82×0.21​​ $t=\sqrt{\frac{0.42}{9.8}}t\; =\; \backslash sqrt\{\backslash frac\{0.42\}\{9.8\}\}$t=9.80.42​​ $t=\sqrt{0.04286}t\; =\; \backslash sqrt\{0.04286\}$t=0.04286​ $t\approx 0.207\hspace{0.17em}secondst\; \backslash approx\; 0.207\; \backslash ,\; \backslash text\{seconds\}$t≈0.207seconds

So, the reaction time of the ruler, given the distance it traveled, is approximately 0.207 seconds.

3. Real-World Example

This estimation process is similar to practical experiments used in physics education, such as timing the fall of an object to determine gravity’s effect. For instance, in a physics lab experiment with a drop timer or reaction time tester, students use similar calculations to understand and measure reaction times and free-fall physics.

Conclusion

In this case, the reaction time, given that the ruler falls 21 cm, is approximately 0.207 seconds. This estimation assumes ideal conditions with no air resistance or other external factors affecting the fall.