A meterstick is held vertically above your hand, with the lower end between your thumb and first finger. On seeing the meter stick released, you grab it with those two fingers. You can calculate your reaction time from the distance the meterstick falls, read directly from the point where your fingers grabbed it.

Derive a relationship for your reaction time in terms of this measured distance d.

d = (g/2)t^2

t = sqrt(2d/g)

To derive a relationship for your reaction time in terms of the measured distance d, we can use the equations of motion for a falling object and assume that the meterstick falls under the influence of gravity alone.

The equation of motion for the distance fallen (d) by a falling object can be expressed as:

d = 0.5 * g * t^2

where:
d is the distance fallen (measured from the point of release to the point of grab),
g is the acceleration due to gravity (approximately 9.8 m/s^2),
t is the time it takes for the meterstick to fall.

We can rewrite this equation to solve for t:

t^2 = (2 * d) / g

Taking the square root of both sides:

t = √(2 * d / g)

Now we have derived a relationship for the reaction time (t) in terms of the measured distance (d).

To derive a relationship for your reaction time in terms of the measured distance, let's consider the motion of the meterstick as it falls and you grab it.

First, we need to understand the motion of a falling object. The distance it falls can be calculated using the equation of motion:

d = (1/2) * g * t^2

where:
d is the distance fallen
g is the acceleration due to gravity (approximately 9.8 m/s^2)
t is the time it takes to fall

We can rearrange the equation to solve for the time:

t^2 = (2 * d) / g

Now, let's introduce the concept of reaction time. Reaction time is the time it takes for you to respond to a visual stimulus and initiate a physical action. In this case, it is the time it takes for you to grab the falling meterstick after seeing it released.

Let's denote the reaction time as 'tr'. From the moment you see the meterstick released, it takes you 'tr' seconds to initiate the grabbing action. Therefore, the total time it takes for the meterstick to fall and for you to grab it is the sum of the falling time and the reaction time: t_total = t + tr.

Now, let's substitute t from the first equation into the above equation for t_total:

t_total^2 = (2 * d) / g + tr^2

Since we are interested in solving for the reaction time 'tr', we can rearrange the equation:

tr^2 = t_total^2 - (2 * d) / g

Taking the square root of both sides gives us the final relationship for the reaction time 'tr' in terms of the measured distance 'd':

tr = √(t_total^2 - (2 * d) / g)

So, by measuring the distance 'd' from the point where your fingers grab the meterstick, you can calculate your reaction time using the derived relationship above.