Example

Question:

Reaction time: When a situation demands our immediate action, it takes some time before we really respond. Reaction time is the time a person takes to observe, think, and act.
You can measure your reaction time by a simple experiment: Ask your friend to drop a ruler vertically through the gap between your thumb and forefinger. After you catch it, find the distance \(d\) travelled by the ruler. In a particular case, \(d = 21.0\,\mathrm{cm}\). Estimate the reaction time.

Solution:

The ruler drops under free fall.
Initial velocity: \(v_0 = 0\), acceleration: \(a = -g = -9.8\,\mathrm{m/s^2}\)
The distance \(d\) and reaction time \(t_r\) are related by:
\[ d = -\frac{1}{2}gt_r^2 \] Taking positive \(d\) (downward motion),
\[ t_r = \sqrt{\frac{2d}{g}} \] Given: \(d = 21.0\,\mathrm{cm} = 0.21\,\mathrm{m}\), \(g = 9.8\,\mathrm{m/s^2}\) \[ t_r = \sqrt{\frac{2 \times 0.21}{9.8}} = \sqrt{\frac{0.42}{9.8}} \approx \sqrt{0.04286} \approx 0.2\,\mathrm{s} \] The estimated reaction time is \(0.2\,\mathrm{s}\).