Question

The speeds of the Fiat and Ferrari racing cars are recorded to $ \pm 4.5\, \times 10{\,^{ - 4}}\,m{\sec ^{ - 1}} $ . Assuming the track distance to be known within $ \pm 16\,m $ , is the uncertainty principle violated for a $ 35000\,kg $ car?

Answer 1

Hint :This is the problem of Heisenberg Uncertainty principle which says that for a particle or we can say for an object it is impossible to calculate position and velocity at the same time so always there is uncertainty between the position and velocity values. If we know about the position means if we are certain about the position then always there is uncertainty in velocity and vice versa. Put the values in the principle.

Complete Step By Step Answer:
The question discusses the speeds of two racing cars one is Faint and second is Ferrari both move with high speed and are fast. It was assumed that the distance of track is $ \pm 16\,m $ it means for those objects we have values of distance. Now let’s try to put the values in formula, we know that Heisenberg uncertainty principle can be written in mathematical form as,
 $ \Delta x\, \times \Delta p\, = \,\dfrac{h}{{4\pi }} $
Where (x) is the position and (p) is the momentum. If we open the formula of momentum which is mass multiplied by velocity then we will get expression like this,
 $ \Delta x\, \times \Delta (mv)\, \geqslant \,\dfrac{h}{{4\pi }} $
Taking mass which is constant for car on the right hand side then, we will get $ \Delta x\, \times \Delta v\, \geqslant \dfrac{h}{{4\pi m}} $
This the relation of uncertainty of position and velocity expression. Now putting value in the formula as- $ \Delta x\, \times \Delta v\, = \,4.5 \times {10^{ - 4}}\, \times 16\, = \,7.2\, \times {10^{ - 3}}\,{m^2}{\sec ^{ - 1}} $
This value should be greater than and equal to $ \dfrac{h}{{4\pi m}} $ now also find out the value of $ \dfrac{h}{{4\pi m}} $ .
 $ \dfrac{h}{{4\pi m}} = \,\dfrac{{6.626\, \times 10{\,^{ - 34}}}}{{4 \times 3.14 \times 3500}} $ = $ 1.507\, \times {10^{ - 38}} $
Since, $ \Delta x\, \times \Delta v\, \geqslant \,\dfrac{h}{{4\pi m}} $ hence Heisenberg Uncertainty principle is not violated.

Note :
The expression can be written for time, here as you see the momentum is given in the principle and that on opening the formula gives mass and velocity. We know mass is constant for cars; it does not change whether we are finding position or velocity, hence we take it to the other side of the equation. The Heisenberg principle can be used to find out the chances of rain tomorrow, it is also called a state of uncertainty.