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Two students want to measure the acceleration a of a cart

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 20SE Chapter 3

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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Problem 20SE

Two students want to measure the acceleration a of a cart rolling down an inclined plane. The cart starts at rest and travels a distance s down the plane. The first student estimates the acceleration by measuring the instantaneous velocity u just as the cart has traveled s meters, and uses the formula a = v2/2s. The second student estimates the acceleration by measuring the time, in seconds, that the cart takes to travel s meters, and uses the formula a = 2s/t2. Assume that s = 1 m, and that there is negligible uncertainty in s. Assume that v = 3.2 ± 0.1 m/s and that t = 0.63 ± 0.01 s. Assume that the measurements of v and t are independent.

a. Compute the acceleration using the method of the first student. Call this estimate a1 Find the uncertainty in a1.

b. Compute the acceleration using the method of the second student. Call this estimate a2. Find the uncertainty in a2.

c. Find the weighted average of a1 and a2 that has the smallest uncertainty. Find the uncertainty in this weighted average.

Step-by-Step Solution:

Answer :

Step 1 of 4:

               Given that, the two students want to measure the acceleration a of a cart rolling down an inclined plane. The cart starts at rest and travels at distance s down the plane.

          The first student estimates the acceleration by measuring the instantaneous velocity u just as the cart has travelled s meters and the formula is used by

                                                 

                               Here, s = 1 m

         The second student estimates the acceleration by measuring the time (in seconds), that the cart takes to travel s meters and the formula is used by

                                         

   Given that  and

     Here are independent.

 Step 2 of 4:

  a). To compute the acceleration using the method of the first student. Let  is this estimate and to find the uncertainty in

                           The acceleration is defined as

                                           

                                                     (here v = 3.2 , s = 1)

                                           

                   Therefore, the acceleration is .

              Now, differentiate with respect to ‘v’, we get

                       

                                =

                                =

                   Now, the uncertainty of is  

              

              =

                        =

                        = (3.2)(0.1)

                = 0.32 .

Conclusion:

            Therefore, the estimated value of the acceleration using the method of the first student is

                                   

                    Here, 0.32 is the uncertainty of .

   

Step 3 of 4

Chapter 3, Problem 20SE is Solved
Step 4 of 4

Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

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