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.

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 .