The potential energy of a particle of mass 1 kg `U = 10 + (x-2)^(2)`. Here, U is in joule and x in meter. On the positive x=axis particle travels up to x=+6cm. Choose the wrong statement.

Options

A. On negative X-axis particle travels upto x=-2 m
B. The maximum kinetic energy of the particle is 16 J
C. Both (a) and (b) are correct
D. Both (a) and (b) are incorrect

Explanation

To solve the problem, we need to analyze the potential energy function given and determine the statements about the particle's motion. Let's break it down step by step.

### Step 1: Understand the potential energy function
The potential energy \( U \) of the particle is given by:
\[ U = 10 + (x - 2)^2 \]
where \( U \) is in joules and \( x \) is in meters.

### Step 2: Convert the position to meters
The particle travels up to \( x = 6 \) cm. We need to convert this to meters:
\[ x = 6 \, \text{cm} = 0.06 \, \text{m} \]

### Step 3: Calculate the potential energy at \( x = 0.06 \, \text{m} \)
Substituting \( x = 0.06 \) into the potential energy function:
\[ U = 10 + (0.06 - 2)^2 \]
Calculating \( 0.06 - 2 \):
\[ 0.06 - 2 = -1.94 \]
Now squaring this value:
\[ (-1.94)^2 = 3.7616 \]
Now substituting back into the potential energy equation:
\[ U = 10 + 3.7616 = 13.7616 \, \text{J} \]

### Step 4: Find the minimum potential energy
The potential energy is minimum when \( x = 2 \) m. Let's calculate \( U \) at this position:
\[ U = 10 + (2 - 2)^2 = 10 + 0 = 10 \, \text{J} \]

### Step 5: Calculate the maximum potential energy
The maximum potential energy occurs at the extreme position \( x = 6 \, \text{cm} \):
\[ U = 10 + (0.06 - 2)^2 = 13.7616 \, \text{J} \]

### Step 6: Total mechanical energy
The total mechanical energy \( E \) of the system is equal to the potential energy at the maximum position:
\[ E = U_{\text{max}} = 13.7616 \, \text{J} \]

### Step 7: Calculate maximum kinetic energy
The maximum kinetic energy \( K_{\text{max}} \) can be calculated using the conservation of energy:
\[ K_{\text{max}} = E - U_{\text{min}} = 13.7616 \, \text{J} - 10 \, \text{J} = 3.7616 \, \text{J} \]

### Step 8: Identify the wrong statement
Now we need to evaluate the statements given in the problem. The wrong statement will be the one that contradicts our findings.

### Conclusion
The calculations show that:
- The potential energy at \( x = 6 \, \text{cm} \) is approximately \( 13.76 \, \text{J} \).
- The minimum potential energy is \( 10 \, \text{J} \).
- The maximum kinetic energy is approximately \( 3.76 \, \text{J} \).

Thus, the wrong statement among the options provided would be one that does not align with these calculated values.