Assume that you are travelling at a speed of 90 km/h in a small car with a mass of 1050 kg. If the uncertainty in the velocity of the car is 1% `(Deltav =0.9 km per h)`, what is the uncertainty (in meters) in the position of the car ?

Options

A. ` Delta x ge 1 xx 10 ^( -35 ) m `
B. ` Delta x ge 2 xx 10^( -37 ) ` m
C. ` Delta x ge 2 xx 10 ^( -36 ) m`
D. `Delta x ge 4 xx 10 ^( - 38 ) ` m

Explanation

To solve the problem, we will use the Heisenberg Uncertainty Principle, which states that the uncertainty in position (Δx) multiplied by the uncertainty in momentum (Δp) is greater than or equal to a constant (h/4π). The momentum (p) can be expressed as the product of mass (m) and velocity (v), so the uncertainty in momentum can be expressed as:

Δp = m * Δv

Where:
- Δv is the uncertainty in velocity.
- m is the mass of the car.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Mass of the car (m) = 1050 kg
- Velocity of the car (v) = 90 km/h
- Uncertainty in velocity (Δv) = 1% of 90 km/h = 0.9 km/h

2. **Convert Δv to SI Units:**
- To convert km/h to m/s, we use the conversion factor: 1 km/h = 1/3.6 m/s.
- Therefore, Δv = 0.9 km/h = 0.9 / 3.6 m/s = 0.25 m/s.

3. **Calculate the Uncertainty in Momentum (Δp):**
- Δp = m * Δv
- Δp = 1050 kg * 0.25 m/s = 262.5 kg·m/s.

4. **Apply Heisenberg Uncertainty Principle:**
- According to the principle: Δx * Δp ≥ h / (4π)
- Where h (Planck's constant) = 6.626 × 10^-34 Js.

5. **Rearranging for Δx:**
- Δx ≥ h / (4π * Δp)
- Δx ≥ (6.626 × 10^-34 Js) / (4 * π * 262.5 kg·m/s).

6. **Calculate Δx:**
- First, calculate the denominator:
- 4 * π * 262.5 ≈ 3295.4.
- Now calculate Δx:
- Δx ≥ (6.626 × 10^-34) / 3295.4 ≈ 2.01 × 10^-37 m.

7. **Final Result:**
- The uncertainty in the position of the car (Δx) is approximately 2.01 × 10^-37 meters.

### Conclusion:
The uncertainty in the position of the car is approximately **2 × 10^-37 meters**.