PH LAB please show analysis and question 1-9?detailed? Kinematics Objeetive: To

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PH LAB

please show analysis and question 1-9?detailed?

Kinematics Objeetive: To describe the motion of an object under constant velocity or constant acceleration graphically and mathematically Theory: In constant velocity motion how does the position vary with time? In motion under a constant acceleration how does the velocity vary with time? How does the position vary wi time under a constant acceleration? If you have a position versus time or velocity versus time graph, how can you take derivatives or integrals using these graphs? Part 1: To describe the motion of an object moving with no net forces acting on it. Procedure: 1) Take a time sequence photo of a cart moving on a horizontal air track. To create the image, have someone open the camera's shutter just after the cart rebounds of the left-hand bumper, and close the shutter just after the cart leaves the field of view. The rotating wheel with 2 slits will create an image on the film every one tenth of a second Record the position of the particle for each time it appears in the photograph. Use the right edge of the ball as the reference point in taking measurements. Use the scale shown in the photograph to make 2) 3) Record the time at which the particle reaches each position in the photograph, and assume the first image is created at time t Osec. Analysis and Questions: 1). Plot position versus time, perform linear regression, and plot your best-fit line. Note 2) Using the regression results, write an equation which describes the position of the 3) Calculate the average velocity of the particle during each time interval using an the physical significance of the slope and intercept. particle as a function of time. interval halving technique. To do this, use the position and time just before and after the time interval of interest. 1+1 4) Calculate the overall average velocity from the interval-halving results. 5) Plot the average velocity of the particle versus time, and draw in the curve which best describes the data. 6) Perform an integral on the velocity curve from f to 6) Perform an integral on the velocity curve from to seconds, by computing the seconds, by computing the area under the curve. What is the physical significance of this area? 7) Calculate the distance traveled for the same time interval 8) Compare results using a percent difference. an equation which describes the velocity of the particle as a function o

Explanation / Answer

I couldn't upload a photo. So have a look at this. This is the basic plot of the data.

Now to perform linear regression, we know:

1. Regression Equation(y) = a + bx

Slope(b) = (NXY - (X)(Y)) / (NX2- (X)2)

Intercept(a) = (Y - b(X)) / N

Here (as default), x is the time and y is the position as its function. N is the total number of datapoints and a and b are the coeefficients to be determined. Put in all the values here and you get:

a = -55.941212412468

b = 151.39777564191

Now put in these values and you get the final plotted function as : -55.941212412468+151.39777564191x shown as:

https://drive.google.com/file/d/0B8D36VXsXeZvY2tWdEw1T0ttU2s/view?usp=sharing

2. So using the result, you get the equation of motion as:

x(t) = 151.39777564191*t - 55.941212412468

3. As given in the formula, you can straightaway use it in an excel worksheet and find the ratio for each pair of two datapoints. The table you will get will be:

0.53   25   130
0.58   31.5 155.5556
0.625 38.5 155
0.665 44.7 140
0.71   51 162.5
0.75   57.5 160
0.8 65.5 150
0.87   76 ------

These will be the aveage speed for the two upper datapoints each time.

4. Halving the intervals, you take the two datapoints so as to give the average speed as:

v = (44.7-25)/(0.665-0.53) = 145.9259259 in the first half

and similarly v = 156.25 in the last half

5. You can see the plot here:

https://drive.google.com/file/d/0B8D36VXsXeZvbWR1TExmeERveVk/view?usp=sharing

You can see the last datapoint where the speed is 150 cm/s matches best.

6. You can integrate the area below the curve by using the area of rectangle trianlge and trapezoid as 0.5 (sum of parallel sides)*(distance between them) and find the area to be as:

In short, you can also find the distance travlled by mutiplying the average speed with the corresponding time interval so that you get the following distances:

0.53   6.5
0.58   6.99999
0.625   6.2
0.665   6.3
0.71   6.5
0.75   8
0.8   10.5
0.87   10.5

This gives a total distance of 61.499 cm.

7. For the same time inteval using the formula found above, we get:

-55.941212412468 + 151.39777564191 * 0.6364 ( the time you gave in the end) = 60.74855

8. Percent difference between the two is:

(60.7485568392168-61.499)/60.74855683921689 = 1.2% approximately

9. Since we use linear regression and we do not have any other data, so we have to assume that the cart is travelling at uniform speed and hence the equation for its velocity will be:

v = 151.39777564191 cm/s

I hope the answers are clear. Please write or comment back in case any doubt. I hope this was helpful.

Thanks

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