The uncertainties in the velocities of two particles, A and B are 0.05 and 0.02 `ms^(-1)`, respectively. The mass of B is five times to that of the mass A. What is the ratio of uncertainties `((Delta_(X_A))/(Delta_(X_B)))` in their positions

Options

A. 2
B. 0.25
C. 4
D. 1

Answer

To solve the problem, we will use the Heisenberg Uncertainty Principle, which states that the uncertainty in position (Δx) and the uncertainty in momentum (Δp) of a particle are related by the equation:

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

where \(h\) is Planck's constant.

### Step 1: Understand the relationship between momentum and velocity
The momentum \(p\) of a particle is given by:

\[ p = mv \]

where \(m\) is the mass and \(v\) is the velocity. The uncertainty in momentum (Δp) can be expressed as:

\[ \Delta p = m \Delta v \]

where Δv is the uncertainty in velocity.

### Step 2: Write the equations for particles A and B
For particle A, the uncertainty in position (Δx_A) can be expressed as:

\[ \Delta x_A = \frac{h}{4\pi m_A \Delta v_A} \]

For particle B, the uncertainty in position (Δx_B) can be expressed as:

\[ \Delta x_B = \frac{h}{4\pi m_B \Delta v_B} \]

### Step 3: Substitute the mass of B in terms of A
Given that the mass of B is five times that of A:

\[ m_B = 5m_A \]

### Step 4: Substitute the expressions for Δx_A and Δx_B
Now we can substitute \(m_B\) into the equation for Δx_B:

\[ \Delta x_B = \frac{h}{4\pi (5m_A) \Delta v_B} \]

### Step 5: Find the ratio of uncertainties in positions
Now, we can find the ratio of the uncertainties in positions:

\[ \frac{\Delta x_A}{\Delta x_B} = \frac{\frac{h}{4\pi m_A \Delta v_A}}{\frac{h}{4\pi (5m_A) \Delta v_B}} \]

This simplifies to:

\[ \frac{\Delta x_A}{\Delta x_B} = \frac{\Delta v_B \cdot 5}{\Delta v_A} \]

### Step 6: Substitute the values of Δv_A and Δv_B
Given:
- Δv_A = 0.05 ms^(-1)
- Δv_B = 0.02 ms^(-1)

Now substituting these values into the ratio:

\[ \frac{\Delta x_A}{\Delta x_B} = \frac{0.02 \cdot 5}{0.05} \]

### Step 7: Calculate the ratio
Calculating the above expression:

\[ \frac{\Delta x_A}{\Delta x_B} = \frac{0.1}{0.05} = 2 \]

Thus, the ratio of uncertainties in their positions is:

\[ \frac{\Delta x_A}{\Delta x_B} = 2 \]

### Final Answer
The ratio of uncertainties in their positions \(\frac{\Delta x_A}{\Delta x_B}\) is **2:1**.

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