Figure 3-40 shows a simple device for measuring your...

Figure $3-40$ shows a simple device for measuring your reaction time. It consists of a cardboard strip marked with a scale and two large dots. A friend holds the strip vertically, with thumb and forefinger at the dot on the right in Fig. $3-40$. You then position your thumb and forefinger at the other dot (on the left in Fig. 3-40), being careful not to touch the strip. Your friend releases the strip, and you try to pinch it as soon as possible after you see it begin to fall. The mark at the place where you pinch the strip gives your reaction time. (a) How far from the lower dot should you place the $50.0 \mathrm{~ms}$ mark? (b) How much higher should the marks for $100,150,200$, and $250 \mathrm{~ms}$ be? (For example, should the $100 \mathrm{~ms}$ marker be two times as far from the dot as the $50 \mathrm{~ms}$ marker? Can you find any pattern in the answers?)

Key Concepts

Free-Fall Motion

This concept refers to the motion of an object solely under the influence of gravity, without air resistance, where the only force acting is gravitational force. In such a scenario, the object's acceleration is constant and equal to the gravitational acceleration, allowing its position and velocity to be described by specific kinematic equations.

Kinematic Equations

Kinematic equations are mathematical formulas that relate displacement, velocity, acceleration, and time for an object moving with constant acceleration. In the context of free-fall motion, one commonly used equation is d = ½ gt², which directly relates the distance fallen to the square of the elapsed time.

Square Law Relationship

This concept describes the non-linear dependence of distance on time during free-fall. Specifically, because displacement in free-fall follows the equation d = ½ gt², the distance fallen is proportional to the square of the time elapsed. This means that if you double the time, the distance increases by a factor of four, which is crucial for understanding the spacing of markers in a reaction time measurement device.

Reaction Time Measurement

Reaction time measurement in physics involves timing the delay between a stimulus and the response of the observer. Instruments often use physical events, such as the free fall of an object, to provide a measurable distance that corresponds to the elapsed reaction time, utilizing the predictable behavior of free-fall motion.

Transcript

00:01 This problem covers the concept of free fall and under free fall the vertical displacement that is delta y is given by the initial velocity that is v y into the time of fall plus half of the gravitational acceleration into time of fall square so for part a where t equals 50 milliseconds so in that the vertical displacement of the scale is equivalent to the initial speed is zero, so we neglect this term.

00:36 So the vertical displacement is half of the gravitational acceleration, that is 9 .8 meters per second square into time square and time is 50 into 10 minus 3 seconds square.

00:53 Or the vertical displacement or the fall in the scale is equivalent to 0 .0122 meter.

01:03 Or we can see the vertical displacement is equivalent to 1 .22 centimeter.

01:10 So for 50 milliseconds mark the scale reading is 1 .2 centimeter.

01:19 For part b at t equals 100 milliseconds, the vertical displacement that is delta y equals half of 9 .8 meters per second square into time square and that is 0 .1 seconds square or the vertical displacement of the scale is 0 .049 meter or we can say 4 .9 centimeter.

01:58 Now for t equals 150 milliseconds, for t equals 150 milliseconds, the vertical displacement is equivalent to of 9 .8 meters per second square into 0 .150 seconds square.

02:23 Or the scale reading corresponding to that time is equivalent to 0 .11 meter or we can say 11 centimeter.

02:34 At t equals 200 milliseconds, the vertical displacement delta y equals half of 9 .8 meters.

02:44 4 seconds square into 0 .2 seconds square or the vertical displacement of the scale equals 0 .196 meter or we can say 19 .6 centimeter...

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