Two students want to measure the acceleration a of a cart

QUESTION:

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 \(v\) just as the cart has traveled \(s\) meters, and uses the formula \(a=v^{2} / 2 s\). The second student estimates the acceleration by measuring the time, in seconds, that the cart takes to travel \(s\) meters, and uses the formula \(a=2 s / t^{2}\). Assume that s = 1 m, and that there is negligible uncertainty in \(s\). Assume that \(v=3.2\pm0.1\mathrm{\ m}/\mathrm{s}\) and that \(t=0.63\pm0.01\mathrm{\ s}\). Assume that the measurements of \(v\) and \(t\) are independent.

a. Compute the acceleration using the method of the first student. Call this estimate \(a_{1}\). Find the uncertainty in \(a_{1}\).

b. Compute the acceleration using the method of the second student. Call this estimate \(a_{2}\). Find the uncertainty in \(a_{2}\).

c. Find the weighted average of \(a_{1}\) and \(a_{2}\) that has the smallest uncertainty. Find the uncertainty in this weighted average.

Equation Transcription:

Text Transcription:

s

v

a = v^2 / s

a = 2s / t^2

v = 3.2 pm 0.1 m/s

t = 0.63 pm 0.01 s

t

a_1

a_2