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Chapter 2: Problem 41

A meter stick 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 these 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 \(17.6 \mathrm{cm},\) what is the reaction time?

Short Answer

Expert verified
The reaction time is approximately 0.19 seconds when the meter stick falls 17.6 cm.

Step by step solution

01

Understand the Problem

We need to calculate the reaction time based on the distance the meter stick falls before being caught. This involves analyzing the motion of the stick under gravity.
02

Use Physics of Free Fall

The meter stick falls freely under gravity, so we can use the equation of motion for free fall without initial velocity: \[ d = \frac{1}{2} g t^2 \]where \(d\) is the distance fallen, \(g\) (approximately \(9.8 \, \text{m/s}^2\)) is acceleration due to gravity, and \(t\) is the time taken to fall.
03

Solve for Reaction Time

Rearrange the equation to solve for \(t\):\[ t = \sqrt{\frac{2d}{g}} \]This is the formula to calculate the reaction time based on the distance \(d\) the stick has fallen.
04

Convert Distance Units

Convert the given distance from centimeters to meters: \(17.6 \, \text{cm} = 0.176 \, \text{m}\).
05

Substitute and Calculate

Substitute the converted distance into the formula to find the reaction time:\[ t = \sqrt{\frac{2 \times 0.176}{9.8}} \]Compute this to get the reaction time.
06

Final Calculation

Calculate the value:\[ t = \sqrt{\frac{0.352}{9.8}} \approx \sqrt{0.0359184} \approx 0.1895 \, \text{s} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Free Fall Motion
The concept of free fall motion is crucial in understanding how objects behave under the influence of gravity when no other forces are acting on them. When we talk about free fall, we essentially refer to the motion of an object that is only driven by gravity. In our scenario, the meter stick falls freely from rest when it is released. Here are a few important features of free fall motion:

  • All objects in free fall, regardless of their mass, experience the same constant acceleration due to gravity.
  • In the absence of air resistance, the distance an object falls is directly related to the square of the time it has been falling.
  • Free fall begins the moment an object is dropped and continues until it is stopped by something else, like your fingers in this exercise.
To derive the distance fallen during free fall, we make use of the formula: \[ d = \frac{1}{2} g t^2\]where:- \(d\) is the distance fallen,- \(g\) is the acceleration due to gravity,- \(t\) is the time elapsed.The above equation assumes the object starts from rest, meaning its initial velocity is zero.
Physics Problem Solving
Solving physics problems often involves a methodical approach that requires understanding and applying underlying principles. The key to success is in understanding what the problem asks and how the physical laws apply to it. Let's break it down step-by-step:

  • Analyze the Problem: Carefully read the problem to identify the key factors involved. In our situation, we want to connect the reaction time with the distance the stick falls.
  • Identify Equations and Concepts: Recognize which physics concepts apply, such as free fall motion and the involved equations.
  • Rearrange the Equation: To find the reaction time \(t\), rearrange the distance formula to solve for time: \[ t = \sqrt{\frac{2d}{g}} \]
  • Calculate and Convert Units: Ensure all values are in the correct units before substituting numbers into the equation. For example, convert centimeters to meters.
  • Execute the Calculations: Substitute the numbers and perform calculations to arrive at a solution.
Breaking a problem into these steps reduces complexity, making it easier to find solutions confidently.
Acceleration due to Gravity
One of the most fundamental constants in physics is the acceleration due to gravity, denoted as \(g\). It represents how fast an object accelerates when falling freely near the Earth's surface.

Here are some key aspects:
  • The standard value of \(g\) is approximately \(9.8 \, \text{m/s}^2\).
  • It acts equally on all objects regardless of their mass, making it possible to calculate how fast something falls or how far, using the free fall formula.
  • This constant provides a link between the time and distance of a falling object, as seen in the formula for free fall motion.
To find the reaction time in the exercise, we use the value of \(g\) in our calculations. It governs the rate of acceleration and thus determines how quickly the meter stick travels those 17.6 cm (converted to meters) before being caught. Understanding \(g\) helps in accurately computing times and distances in free fall scenarios.

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Most popular questions from this chapter

Launch Failure. A \(7500-\mathrm{kg}\) rocket blasts off vertically from the launch pad with a constant upward acceleration of 2.25 \(\mathrm{m} / \mathrm{s}^{2}\) and feels no approciable air resistance. When it has reached a height of \(525 \mathrm{m},\) its engines suddenly fail so that the only force acting on it is now gravity. (a) What is the maximum height this rocket will reach above the launch pad? (b) How much time after engine failure will elapse before the rocket comes crashing down to the launch pad, and how fast will it be moving just before it crashes? (c) Sketch \(a_{y}-t, v_{y}-t,\) and \(y-t\) graphs of the rocket's motion from the instant of blast-off to the instant just before it strikes the launch pad.

During launches, rockets often discard unneeded parts. A certain rocket starts from rest on the launch pad and accelerates upward at a steady 3.30 \(\mathrm{m} / \mathrm{s}^{2} .\) When it is 235 \(\mathrm{m}\) above the launch pad, it discards a used fuel canister by simply disconnecting it. Once it is disconnected, the only force acting on the canister is gravity (air resistance can be ignored). (a) How high is the rocket when the canister hits the launch pad, assuming that the rocket does not change its acceleration? (b) What total distance did the canister travel between its release and its crash onto the launch pad?

A turtle crawls along a straight line, which we will call the \(x\) -axis with the positive dircction to the right. The equation for the turtle's position as a function of time is \(x(t)=50.0 \mathrm{cm}+\) \((2.00 \mathrm{cm} / \mathrm{s}) t-\left(0.0625 \mathrm{cm} / \mathrm{s}^{2}\right) t^{2},(\mathrm{a})\) Find the turte's initial velocity, initial position, and initial acceleration. (b) At what time \(t\) is the velocity of the turtle zero? (c) How long after starting does it take the turtle to return to its starting point? (d) At what times \(t\) is the turtle a distance of 10.0 \(\mathrm{cm}\) from its starting point? What is the velocity (magnitude and direction) of the turtle at each of these times? (e) Sketch graphs of \(x\) versus \(t, v_{x}\) versus \(t,\) and \(a_{x}\) versus \(t\) for the time interval \(t=0\) to \(t=40 \mathrm{s}\)

The acceleration of a particle is given by \(a_{x}(t)=\) \(-2.00 \mathrm{m} / \mathrm{s}^{2}+\left(3.00 \mathrm{m} / \mathrm{s}^{3}\right) t .\) (a) Find the initial velocity \(v_{\mathrm{ax}}\) such that the particle will have the same \(x\) -coordinate at \(t=4.00 \mathrm{s}\) as it had at \(t=0\) (b) What will be the velocity at \(t=4.00 \mathrm{s} ?\)

A Fast Pitch. The fastest measured pitched baseball left the pitcher's hand at a speed of 45.0 \(\mathrm{m} / \mathrm{s}\) . If the pitcher was in contact with the ball over a distance of 1.50 \(\mathrm{m}\) and produced constant acceleration, (a) what acceleration did he give the ball, and (b) how much time did it take him to pitch it?
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