chapter2

=CHAPTER 2 : MOTION IN ONE-DIMENSION = The study of motion is referred to as **Kinematics** and the simplest form is along a straight line. This is motion in one-dimension. Motion along a straight line. Similar to a car on a straight road.

 There are three things to know about in this chapter__:__ ** Displacement, Velocity ,** and ** Acceleration **
 * Displacement is symbolized with (x)
 * Velocity is symbolized with (v)
 * Acceleration is symbolized with (a)

** Distance :** Total length of travel. SI unit: meter, m ** Displacement :** the change of position x, or the //distance// from the final position and initial position. Displacement = change in position = final position - initial position Displacement = 

The difference between distance and is that distance is the total amount traveled whereas displacement is only concerned about difference between the initial and final positions. //For example://
 * //If you walk 4m forward your displacement is 4m.//
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">//If you start at a position 5m and go to 12m then your displacement is 7m. 12m - 5m = 7m.//
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">//If you start at 5m and travel to 12m and then back to 4m, then your displacement is -1m. 4m - 5m = - 1m. However your total distance would be 15m. 12m - 5m + 12m - 4m = 15m.//

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">** Average Velocity :**the ratio of displacement to the change in time. SI Units: Meters/second, m/s

<span style="color: #ff0000; font-family: 'Times New Roman',Times,serif; font-size: 120%;">Average Velocity = Change of position / Change in time = <span style="font-family: 'Times New Roman',Times,serif;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **Average Speed** : the ratio of the total distance traveled to the change in time. SI Units: Meters/second, m/s

<span style="color: #ff0000; font-family: 'Times New Roman',Times,serif; font-size: 120%;">Average Speed = total distance / total time = x/t

<span style="color: #ff0000; font-family: 'Times New Roman',Times,serif; font-size: 130%;">**Instantaneous Velocity/Acceleration** <span style="font-family: 'Times New Roman',Times,serif;"> <span style="color: #030303; font-family: 'Times New Roman',Times,serif; font-size: 120%;">The average velocity is the slope on the distance versus time graph.

<span style="font-family: 'Times New Roman',Times,serif;"> <span style="color: #030303; font-family: 'Times New Roman',Times,serif; font-size: 120%;">is the slope of x-versus-t graph. This implies that the slope of the line tangent to the x-versus-t curve is the change in position / change in time. As the change in t approaches 0, or at an instant. <span style="font-family: 'Times New Roman',Times,serif;">

<span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;"><span style="background-color: #ffffff; color: #000000; display: block; font-size: 16px; text-align: left; text-decoration: none;">This allows us to write that the derivative of displacement (x) is velocity (v) and the derivative of velocity is acceleration (a).Or, // x '(t) = v (t) ; v '(t) = a (t) // <span style="font-family: 'Times New Roman',Times,serif;">

<span style="color: #ff0000; font-family: 'Times New Roman',Times,serif; font-size: 120%;"> A line's slope may be positive, negative, or zero; consequently, instantaneous velocity (1D motion) may be positive (x increasing), negative (x decreasing), or zero (no motion). In other words, if the velocity is positive it is moving in the positive x direction. If the velocity is negative it is moving a negative x direction.

<span style="color: #ffff00; font-family: 'Times New Roman',Times,serif; font-size: 110%;"> The magnitude of the instantaneous velocity is the ** instantaneous speed. **

<span style="color: #ff0000; font-family: 'Times New Roman',Times,serif; font-size: 170%;">** Acceleration : ** The ratio of change of velocity with time. SI Units: meter per second per second, m/s 2

<span style="color: #008000; font-family: 'Times New Roman',Times,serif; font-size: 130%;">Average Acceleration= <span style="font-family: 'Times New Roman',Times,serif;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> Instantaneous Acceleration : the limit of the ratio change in velocity divided by the change in time as the change in time approcahes zero. On a plot of velocity versus time, the instantaneous acceleration of time //t// is the slope of the line tangent to the curve at that time:

<span style="font-family: 'Times New Roman',Times,serif;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: 130%;">= Slope of the line tangent to the //v-//versus-//t// curve

<span style="color: #008000; font-family: 'Times New Roman',Times,serif; font-size: 130%;">Therefore, acceleration is the derivative of velocity with respect to time, //dv/dt.//

<span style="font-family: 'Times New Roman',Times,serif;"> In a free falling situation an object falls with g ravitational acceleration - The gravitational acceleration of objects near the earth's surface is the same for all objects regardless of mass and is given by the number

<span style="color: #008080; display: block; font-family: 'Times New Roman',Times,serif; font-size: 19px; text-align: left; text-decoration: none;">// g // = - 9.8m/s

<span style="font-family: 'Times New Roman',Times,serif;">** Motion with Constant Acceleration **

<span style="color: #030303; font-family: 'Times New Roman',Times,serif; font-size: 120%;">With constant acceleration relationships of displacement, velocity and acceleration with time (t) are as follows: <span style="color: #030303; font-family: 'Times New Roman',Times,serif; font-size: 120%;">(Know as Kinematics equations)

<span style="color: #030303; font-family: 'Times New Roman',Times,serif; font-size: 16px;">These equations are found through manipulating the previous concepts

<span style="color: #030303; font-family: 'Times New Roman',Times,serif; font-size: 16px;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: 170%;">Applying the Calculus to Motion

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">It is now known that the derivative of x(t) is v(t) and the derivative of v(t) is a(t). This also means the antiderivative of v(t) is x(t) and the antiderivative of a(t) is v(t):

<span style="font-family: 'Times New Roman',Times,serif;"> <span style="font-family: 'Times New Roman',Times,serif;">and <span style="font-family: 'Times New Roman',Times,serif;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Without going into a full lesson of calculus, the basic concept is this <span style="font-family: 'Times New Roman',Times,serif;"> The Fundamental Theorem of Calculus

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">To apply that in a visual sense, it means this. On a velocity versus time graph, the area under t1 and t2 is the displacement x or s. It also means that on a acceleration versus time graph, the area under t1 and t2 is the velocity.

<span style="font-family: 'Times New Roman',Times,serif;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">As the slope of an x-versus-t graph is the velocity, the area of a v-versus-t graph is the displacement. <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">As the slope of an v-versus-t graph is the acceleration, the area of an a-versus-t graph is the velocity

<span style="font-family: 'Times New Roman',Times,serif; font-size: 170%;">Example Problems:

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">1. An object is dropped from rest and falls a distance D in a given time. If the time during which it falls is doubled, the distance it falls will be <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">(a) 4D <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">(b) 2D <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">(c) D <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">(d) D/2 <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">(e) D/4

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Answer: (a) <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">In this problem the object is first dropped a distance D from rest. <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Then when the time is doubled for the drop <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">This makes the total distance the object falls 4x large. or 4D

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">2. A particle moves with velocity v = 8t - 7, where v is in meters per second and t is in seconds. (a) Find the average acceleration for the one-second intervals beginning at t = 3 s and t = 4 s. (b) Sketch v versus t. What is the instantaneous acceleration at any time?

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">(a) Between 3s and 4s there are two one-seconded intervals: one between 3s and 4s and one between 4s and 5s. <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">When t=3s, v=17m/s; when t=4s, v=25m/s; when t=5s, v=33m/s or v(3) = 17 m/s; v(4) = 25 m/s; v(5) = 33 m/s

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Find a av for the two 1-s intervals <span style="font-family: 'Times New Roman',Times,serif;"> <span style="font-family: 'Times New Roman',Times,serif;">

<span style="font-family: 'Times New Roman',Times,serif;">(b)

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">3. An automobile accelerates from rest at 2 m/s2 for 20 s. The speed is then held constant for 20 s, after which there is an acceleration of -3 m/s2 until the automobile stops. What is the total distance traveled?

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Answer: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">1. Determine the distance traveled during first 20 s and the speed at the end of first 20 s <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">2. Find Dx2 = distance covered between 20 s and 40 s <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">3. Find Dx3 = distance during deceleration <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">4. Find total distance

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">v(20) = at = (2 m/s2)(20 s) = 40 m/s <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Dx1 = vavt = (20 m/s)(20 s) = 400 m <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Dx2 = (40 m/s)(20 s) = 800 m <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Dx3 = (40 m/s)2/[2(3 m/s2)] = 267 m <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">x = Dx1 + Dx2 + Dx3 = 1467 m

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">4. An object is dropped from a height H. During the final second of its fall, it traverses a distance of 38 m. What was H?

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Answer: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Let vf be the final speed before impact, vf-1 the speed 1 s before impact. <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">1. Find the average speed in last second <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">2. Express vf-1 in terms of vf and solve for vf <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">3. Use vf to determine H

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">v av (1 s) = 38 m = 1/2(vf + vf-1)(1 s); vf + vf-1 = 76 m/s <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">vf - vf-1 = 9.81 m/s; vf = 42.9 m/s <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">H = (vf^2)/(2g) = 93.8 m

<span style="font-family: 'Times New Roman',Times,serif;">Labortory Experiment

<span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Introduction: <span style="font-family: 'Times New Roman',Times,serif;">In this lab, you will investigate the motion of a rolling cart as it travels in a straight line. Although this setup may seem oversimplified, you will soon see that a detailed understanding of linear motion is at the heart of many important concepts in physics. In this lab, you will also become familiar with the LabPro interface and techniques for graphing position, velocity, and acceleration in real time. Finally, you will practice using differentiation and integration to investigate the relationships between position, velocity, and acceleration.

<span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Materials
 * <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Cart
 * <span style="font-family: 'Times New Roman',Times,serif;">Set of weights
 * <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">A track
 * <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Low-friction pulley
 * <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">LabPro interface with motion detector

<span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Setup and procedure <span style="font-family: 'Times New Roman',Times,serif;">

<span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Your cart and track should be set up and ready to go. The set up should appear similar to the illustration above. All you need to do before starting is get the computer and LabPro interface ready to take data. Everything you need is in the plastic LabPro box.
 * 1) <span style="font-family: 'Times New Roman',Times,serif;">Pull out the LabPro interface. It is teal and about the size of a graphing calculator. Plug the AC adaptor into the LabPro and then into the wall socket.
 * 2) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Locate the motion detector. Use a cable that has two white ends to connect the motion detector to a "DIGI" port on the LabPro. Place the motion detector upright on the table so that the speaker is pointing straight down the track.
 * 3) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Use a USB cable to connect the LabPro to the computer.
 * 4) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">You are now ready to take data. Click on the green "collect" button at the upper right corner of the screen. This should make the motion detector start to click. If this is not working, ask for help from your instructor. Move the cart around while "collecting" to make sure that the motion detector is working properly.
 * 5) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">The motion detector has a limited range. If your cart gets too close or too far, the detector will not read the correct distance. Move the cart toward the sensor while collecting data until the sensor stops reading the correct distance. Mark this spot on the rulers using blue tape and never use any data that is taken when the cart is beyond this point for the rest of the lab.
 * 6) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Now move the cart as far as you can away from the sensor until it stops reading the correct distance. Mark this spot on the rulers.
 * 1) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Now move the cart as far as you can away from the sensor until it stops reading the correct distance. Mark this spot on the rulers.

<span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Part I – Constant Motion. <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Procedure
 * 1) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Click on the stopwatch button at the upper right corner of the screen. Set the program to collect data for 60 seconds.
 * 2) Choose one team member to move the cart. Start the cart close to the motion detector and move it slowly away from the detector at a <span style="font-family: 'Times New Roman',Times,serif;">constant speed . The constant speed can be obtained by only tapping the cart. (There are no forces on the cart afterward. Friction is negligible.)
 * 3) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">You should watch the screen as you do this and observe the shapes of the graphs. The position vs. time graph should be a straight, slanted line, and the velocity vs. time graph should be a horizontal line.
 * 4) Once you reach the motion sensor’s limit, stop the cart for a few seconds. Then bring the cart slowly back toward the sensor, again at <span style="font-family: 'Times New Roman',Times,serif;">//constant speed.// Repeat for the duration of the thirty seconds. Use File� Save As to save your data with a unique file name.
 * 5) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Repeat steps two-three with a different team member, and move the cart more quickly than before. Again, save your data.

<span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Analysis <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Follow these steps for both of your data sets:
 * 1) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Choose one straight-line segment of the position vs. time graph. Highlight this region.
 * 2) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Use Analyze Linear Fit to fit a straight line to your data. When you perform this step, the computer is finding the equation of line with slope and intercept that best match your data.
 * 3) <span style="font-family: 'Times New Roman',Times,serif;">Write out the equation for position vs. time for constant-velocity motion. Explain why the slope of the linear fit corresponds to velocity.
 * 4) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Highlight the corresponding region of the velocity vs. time graph.
 * 5) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Use Analyze Statistics to find the mean of the velocity
 * 6) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Do your values for velocity from steps 2 and 5 match?
 * 7) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Highlight the corresponding region of the acceleration vs. time graph. Use Analyze Statistics to find the mean of the acceleration
 * 8) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Explain what this value of acceleration means..
 * 9) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Once you have completed the analysis, print up each graph. Also answer the following:
 * 10) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">How did the velocity for the second run compare with that of the first?
 * 11) <span style="font-family: 'Times New Roman',Times,serif;">Did the faster motion correspond to a steeper or shallower slope on the position vs. time graph?

<span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Part II – Accelerated Motion – Speeding Up. Investigating Derivatives. <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Procedure <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Analysis
 * 1) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Use the stopwatch button at the upper right corner of the screen to change the data collection duration to 8 seconds.
 * 2) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Pull the thread that is attached to the cart through the pulley at the end of the table. Place a 500g weight on the cart, and get a 50g weight ready to hang from the loop at the other end of the thread. Place the cart next to the motion sensor.
 * 3) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Have one team member hit "collect." Have another team member place the 50g weight in the loop and release the cart. Repeat until you get a clean graph.
 * 1) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Highlight the good region of data on your graph. You do not want to pay any attention to data that was collected before the cart was released or after the weight hit the ground.
 * 2) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Use Analyze Curve Fit to have the program find the parabola that best matches your data. Choose "Quadratic," then click, "Try Fit." If the curve seems to match your data well, click "OK," otherwise ask your instructor for assistance.
 * 3) <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">Write out the equation for position as a function of time for accelerated motion. Identify how each part of the curve fit relates to this equation, and fill in numbers in the table below.

<span style="font-family: 'Times New Roman',Times,serif;">x(t) = _
 * <span style="font-family: 'Times New Roman',Times,serif;">a ||>  ||
 * <span style="font-family: 'Times New Roman',Times,serif;">v0 ||>  ||
 * <span style="font-family: 'Times New Roman',Times,serif;">x0 ||>  ||

<span style="font-family: 'Times New Roman',Times,serif;">Example Data: <span style="font-family: 'Times New Roman',Times,serif;"> <span style="font-family: 'Times New Roman',Times,serif;"> <span style="font-family: 'Times New Roman',Times,serif;">

<span style="font-family: 'Times New Roman',Times,serif;">Lab: []

<span style="font-family: 'Times New Roman',Times,serif; font-size: 150%;">Learning motion in one dimension is key to a multitude of physics. Cars, humans,particles, and most objects can move. Understanding their motions in one dimension lets us explore more than just one dimension but two dimension, and various factors. It is the foundation of mechanics in physics.

<span style="font-family: 'Times New Roman',Times,serif;">All images are linked to their sources.


 * CATEGORY ||  SCORE (1-4)  ||  POINTS (0-20)  ||
 * Content || 4 || 20 ||
 * Organization || 4 || 20 ||
 * Accuracy || 4 || 20 ||
 * Appearance || 4 || 20 ||
 * Participation || 4 || 20 ||
 * TOTAL || 100 ||