Question

Calculate the uncertainty in velocity of a cricket ball of mass $150g$ if the uncertainty in its position is of the order of $1\overset{{}^\circ }{\mathop{A}}\,(h=6.6\times {{10}^{-34}}kg{{m}^{2}}{{s}^{-1}})$ .

Answer 1

Hint: Before talking about the answer, you should know about the Heisenberg uncertainty principle, which states that we cannot determine the exact position and momentum of a particle simultaneously. The value of Planck’s constant is $6.626\times {{10}^{-34}}$.

Formula used:\[\Delta x\Delta p\ge \dfrac{h}{4\pi }\]
\[\Rightarrow \Delta x.m\Delta v\ge \dfrac{h}{4\pi }\]
where, $\Delta x$ is the uncertainty in position, $m$ is the mass, $v$ is the velocity and $h$ is the Planck’s constant $h=6.626\times {{10}^{-34}}$ and $\pi =3.14$ .
where, $p$ is the momentum, $m$ is the mass and $v$ is the velocity.

Complete step by step answer:
Here, it is given that the mass of the cricket ball is $150g$ and uncertainty in position is $1\overset{{}^\circ }{\mathop{A}}\,$.
The formula used for Heisenberg uncertainty principle is:
$\Delta x\Delta p\ge \dfrac{h}{4\pi }$ – (1)
And $\Delta p=m\Delta V$ – (2)
where, $p$ is the momentum, $m$ is the mass and $v$ is the velocity.
Now, substituting the value of $\Delta p$ in equation (1), we get,
$\Delta x.m\Delta v\ge \dfrac{h}{4\pi }$
where, $\Delta x$ is the uncertainty in position, $m$ is the mass, $v$ is the velocity and $h$ is the Planck’s constant $h=6.626\times {{10}^{-34}}$ and $\pi =3.14$ .
Now, substituting the values in the above formula, we get,
\dfrac${{10}^{-10}}\times 0.15\times \Delta v=\dfrac{6.626\times {{10}^{-34}}}{4\times 3.14}$
 $\Rightarrow \Delta v=3.5\times {{10}^{-24}}m{{s}^{-1}}$

Therefore, the uncertainty in the velocity of cricket ball is \[3.5\times {{10}^{-24}}m{{s}^{-1}}\] .

Additional information:
Heisenberg’s uncertainty principle is a very important part of quantum mechanics. We know where the particle is located then, we know nothing about its momentum. It states that it is impossible to calculate the position and momentum of a particle simultaneously with arbitrarily high precision.
Planck’s constant is defined as a fundamental physical constant that relates photon energy to its frequency. It is denoted with the symbol $'h'$ .
$h=6.626\times {{10}^{-34}}J\,s$
Momentum is a vector quantity which can be calculated from the product of mass and velocity. Its SI unit is kilogram meter per second $(kg\,m/s)$ .

Note: The exact position and momentum cannot be determined simultaneously because electrons do not have direction of motion and definite position at the same time. Momentum is defined as the product of mass and velocity.