KVPY Sample Paper KVPY Stream-SX Model Paper-14

A measurement establishes the position of a proton with an accuracy of \[\pm \,1.00\times {{10}^{-\,11}}m.\] Uncertainly in the proton's position 1 s later is (take,\[\upsilon <<c\])

A) 3.15 mm

B) 3.15 cm

C) 3.15 m

D) 3.15 km

Correct Answer: D

Solution :

Let \[\Delta {{x}_{0}}=\]uncertainly in position at time \[t=0\,s,\]so uncertainly in momentum is \[\Delta p\ge \frac{h}{4\pi \Delta {{x}_{0}}}\]

As \[v<<c,\]momentum uncertainly is \[\Delta p=\Delta (mv)={{m}_{0}}\Delta v;\]

Also; the uncertainly in proton's velocity is \[\Delta v=\frac{\Delta p}{{{m}_{0}}}=\frac{h}{4\pi {{m}_{0}}\Delta {{x}_{0}}}\]

So, the distance x that proton covers in time t cannot be known more accurately than

\[\Delta x=t\Delta v\ge \frac{ht}{4\pi {{m}_{0}}\Delta {{x}_{0}}}\]

At \[t=1s,\]value of \[\Delta x\]is

\[\Delta x\ge \frac{1.054\times {{10}^{-\,34}}\times 1}{2(1.672\times {{10}^{-\,27}})\times (1\times {{10}^{-\,11}})}\]\[\ge 3.15\times {{10}^{3}}m\]

or \[\Delta x\ge 3.15\,km\]