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Step-by-Step Solution
Step 1: State Heisenberg's Uncertainty Principle
According to Heisenberg's Uncertainty Principle:
$$
\Delta x \cdot \Delta p \ge \frac{h}{4 \pi}
$$
where
$ \Delta x $ is the uncertainty in position,
$ \Delta p $ is the uncertainty in momentum, and
$ h $ is Planck’s constant.
Step 2: Express Momentum Uncertainty in Terms of Velocity
Momentum $ p $ is the product of mass ($ m $) and velocity ($ v $). Thus,
$$
\Delta p = m \cdot \Delta v.
$$
Substituting $ \Delta p = m \, \Delta v $ into Heisenberg's Uncertainty Principle gives:
$$
\Delta x \cdot m \cdot \Delta v \ge \frac{h}{4 \pi}.
$$
Step 3: Rearrange the Formula to Find $ \Delta v $
Solving for $ \Delta v $,
$$
\Delta v \ge \frac{h}{4 \pi \, m \, \Delta x}.
$$
Step 4: Substitute the Given Values
Given:
Mass of the particle, $ m = 25 \text{ g} = 0.025 \text{ kg}$
Uncertainty in position, $ \Delta x = 10^{-5} \text{ m}$
Planck's constant, $ h = 6.6 \times 10^{-34} \text{ J s}$
Now substitute these values into the formula:
$$
\Delta v = \frac{6.6 \times 10^{-34}}{4 \times \pi \times 0.025 \times 10^{-5}}.
$$
Step 5: Numerically Evaluate the Expression
Let's approximate $ \pi \approx 3.14 $:
$$
\Delta v \approx \frac{6.6 \times 10^{-34}}{4 \times 3.14 \times 0.025 \times 10^{-5}}.
$$
Carrying out the calculations yields:
$$
\Delta v \approx 2.1 \times 10^{-28} \, \text{m s}^{-1}.
$$
Final Answer
The uncertainty in the velocity of the particle is
$$ 2.1 \times 10^{-28} \text{ m s}^{-1}. $$
