Question

A meter stick is held vertically above your hand, with the lower end between your thumb and first finger. When you see the meter stick released, you grab it with those two fingers. You can calculate your reaction time from the distance the meter stick falls, read directly from the point where your fingers grabbed it. (a) Derive a relationship for your reaction time in terms of this measured distance, \(d\). (b) If the measured distance is 17.6 cm, what is your reaction time?

Short answer

(a) \(t = \sqrt{\frac{2d}{g}}\); (b) the reaction time is approximately 0.19 seconds.

Step-by-step solution

Understand the Motion

The meter stick is in free fall until you catch it, so it undergoes uniform acceleration due to gravity. The relevant equation for the motion is the kinematic equation: \[ d = \frac{1}{2} g t^2 \] where \(d\) is the distance fallen, \(g\) is the acceleration due to gravity (approximately \(9.8\, \text{m/s}^2\) on the surface of Earth), and \(t\) is the time it takes for the stick to fall (your reaction time).

Solve for Reaction Time

We need to solve the kinematic equation for the reaction time \(t\). Starting with \[ d = \frac{1}{2} g t^2 \]we can rearrange to find \(t\):\[ t^2 = \frac{2d}{g} \]Taking the square root of both sides gives us:\[ t = \sqrt{\frac{2d}{g}} \]

Substitute Known Values

Now, substitute the known values into the equation. The distance \(d\) is given as \(17.6\, \text{cm}\), which is \(0.176\, \text{m}\), and \(g = 9.8\, \text{m/s}^2\). Substitute these values into the equation for \(t\):\[ t = \sqrt{\frac{2 \times 0.176}{9.8}} \]

Calculate

Calculate the value of the reaction time:\[ t = \sqrt{\frac{0.352}{9.8}} \approx \sqrt{0.0359} \approx 0.1895 \, \text{seconds} \]

Conclusion

The reaction time when the meter stick falls 17.6 cm before being caught is approximately \(0.19\, \text{seconds}\).

Key concepts

Kinematic Equations

Kinematic equations are fundamental tools in physics that describe the motion of objects. They connect an object's displacement, velocity, acceleration, and time. These equations are especially useful in scenarios where an object moves with constant acceleration. In our example of a meter stick in free fall, we use these equations to calculate reaction time.
A core kinematic equation used here is:

• \[ d = \frac{1}{2} g t^2 \]

This relates the distance (\(d\) ) the meter stick falls to the time (\(t\)) it takes, incorporating the acceleration due to gravity (\(g\)). By knowing the distance an object falls, we can solve for the time, which corresponds to your reaction time in this scenario. This is possible because the distances fallen in free fall scenarios can be calculated by understanding how gravity acts on the object over time.
To find the reaction time, rearrange the kinematic equation to solve for \(t\):

• \[ t = \sqrt{\frac{2d}{g}} \]

Here, you simply plug in your known values of \(d\) and \(g\) to get \(t\).

Free Fall Motion

Free fall motion describes any motion of an object where gravity is the only force acting upon it. This means there's no air resistance or other force involved. In the exercise, the meter stick experiences free fall from the moment it is released until you catch it. During free fall, all objects accelerate at the same rate, regardless of their mass. Provided they are near the Earth's surface, they will accelerate at approximately \(9.8 \, \text{m/s}^2 \).
To fully grasp the motion of a falling object, such as our meter stick, it's important to consider the initial conditions. When you release the meter stick, its initial velocity is zero. The only influence on its motion is gravity, which increases its velocity as it falls. The further it falls before being caught, the more time has elapsed, allowing for a direct measure of reaction time.
This behavior is elegantly modeled by kinematic equations, allowing us to use measured distances to calculate reaction times. With a free-falling object, knowing just the distance fallen and the acceleration due to gravity is sufficient to compute the time of fall.

Acceleration due to Gravity

Acceleration due to gravity is the rate at which any object accelerates when it's in free fall at the Earth's surface. It's typically approximated as \(9.8 \, \text{m/s}^2 \). This acceleration is a constant, which means every second, the velocity of a freely falling object increases by \(9.8 \, \text{m/s}\), provided the object is falling directly downwards without any interference from other forces.
Gravity is a force that attracts two bodies towards each other, and on Earth, it gives mass weight. It acts as a central player in our exercise, the primary force that causes the meter stick to fall when released. Without this predictable acceleration, calculating things like reaction time from fallen objects would not be feasible.
When performing calculations involving gravity, it is crucial to ensure that other forces, such as air resistance, are negligible. This lets us assume that gravitational acceleration remains constant, simplifying our calculations significantly. In practical terms, this constant acceleration allows us to predict and understand the movements of falling objects, and when combined with kinematic equations, it provides a reliable way to solve for unknown variables like time or distance.