Calculate the uncertainty in the velocity of al wagon of mass 2000 kg whose positions is known to an accuracy of `+-10m`.

Answer

To calculate the uncertainty in the velocity of a wagon of mass 2000 kg, whose position is known to an accuracy of ±10 m, we will use Heisenberg's Uncertainty Principle. The principle states that the uncertainty in position (Δx) and the uncertainty in momentum (Δp) are related by the equation:

\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]

Where:
- \( \Delta x \) is the uncertainty in position,
- \( \Delta p \) is the uncertainty in momentum,
- \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \) J·s.

### Step-by-Step Solution:

**Step 1: Identify the values.**
- Mass of the wagon (m) = 2000 kg
- Uncertainty in position (Δx) = ±10 m

**Step 2: Calculate the uncertainty in momentum (Δp).**
The uncertainty in momentum can be expressed as:
\[ \Delta p = m \cdot \Delta v \]
where Δv is the uncertainty in velocity.

**Step 3: Substitute Δp into Heisenberg's equation.**
From Heisenberg's Uncertainty Principle:
\[ \Delta x \cdot (m \cdot \Delta v) \geq \frac{h}{4\pi} \]

**Step 4: Rearrange the equation to solve for Δv.**
\[ \Delta v \geq \frac{h}{4\pi \cdot m \cdot \Delta x} \]

**Step 5: Substitute the known values into the equation.**
- \( h = 6.626 \times 10^{-34} \) J·s
- \( m = 2000 \) kg
- \( \Delta x = 10 \) m

\[ \Delta v \geq \frac{6.626 \times 10^{-34}}{4 \cdot \pi \cdot 2000 \cdot 10} \]

**Step 6: Calculate the denominator.**
\[ 4 \cdot \pi \cdot 2000 \cdot 10 = 4 \cdot 3.14 \cdot 20000 = 251200 \]

**Step 7: Calculate Δv.**
\[ \Delta v \geq \frac{6.626 \times 10^{-34}}{251200} \approx 2.64 \times 10^{-39} \text{ m/s} \]

### Final Result:
The uncertainty in the velocity of the wagon is approximately:
\[ \Delta v \approx 2.64 \times 10^{-39} \text{ m/s} \]

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