Question

Table-tennis ball has a mass 10 g and a speed of 100 m/s. If speed can be measured within an accuracy of 10 %, what will be the uncertainty in speed and position respectively ?
a.) 10, 4$\times {10^{ - 33}}$
b.) 10, 5.27$\times {10^{ - 34}}$
c.) 0.1, 5$\times {10^{ - 34}}$
d.) None of these

Answer 1

The uncertainty in speed and position can be found by the Heisenberg’s formula-
$\Delta x\Delta p$= $\dfrac{h}{{4\Pi }}$
Where $\Delta x$= uncertainty in position
$\Delta v$= uncertainty in velocity
‘h’ = Planck’s constant

Complete answer:
First, we will write what is given to us and what we want to find.
Given :
Mass of tennis ball = 10 g
Speed of tennis ball = 100 m/s
Accuracy of measurement = 10 %
To find :
Uncertainty in speed and position
We know that according to Heisenberg’s uncertainty principles
$\Delta x\Delta p$= $\dfrac{h}{{4\Pi }}$
$\Delta x\Delta (mv)$=$\dfrac{h}{{4\Pi }}$
$\Delta x \cdot m\Delta v$= $\dfrac{h}{{4\Pi }}$
We have the accuracy of measurement = 10 %
And the speed of the tennis ball = 100 m/s
Thus, $\Delta v$= $\dfrac{{10}}{{100}} \times$v
$\Delta v$= $\pm$10 m/s
Filling the values, we have
$\Delta x \times {10^{ - 2}} \times 10$= $\dfrac{{6.626 \times {{10}^{ - 34}}}}{{4 \times 3.14}}$
On solving, we get
$\Delta x$= 5.27$\times {10^{ - 34}}$m
So, the uncertainty in speed is 10 m/s and the position is 5.27$\times {10^{ - 34}}$m.

Thus, option b.) is the correct answer.

Note:
It must be noted that we have written $\Delta v$= $\pm$10 m/s. But we have taken speed = 10 m/s. This is because speed is always positive. The velocity can be negative or positive or zero. The speed can be zero or positive.