A particle of mass 4 kg moves along x axis, potential energy ( U ) varies with respect to x as `U = 20 + ( x-4)^(2)` , maximum speed of paritcle is at

Options

A. `x=4`
B. `x=2`
C. ` x=0`
D. `x=2.5`

Answer

To find the position where the particle of mass 4 kg has its maximum speed, we need to analyze the potential energy function given by:

\[ U(x) = 20 + (x - 4)^2 \]

### Step 1: Understand the Potential Energy Function
The potential energy function indicates how the potential energy varies with the position \( x \). The term \( (x - 4)^2 \) suggests that the potential energy reaches its minimum when \( x = 4 \).

### Step 2: Find the Minimum Potential Energy
To find the minimum potential energy, we can evaluate the potential energy function at \( x = 4 \):

\[ U(4) = 20 + (4 - 4)^2 = 20 + 0 = 20 \]

### Step 3: Determine Conditions for Maximum Speed
In mechanics, the total mechanical energy \( E \) of a system is conserved and is the sum of kinetic energy \( K \) and potential energy \( U \):

\[ E = K + U \]

The kinetic energy \( K \) is given by:

\[ K = \frac{1}{2} mv^2 \]

For maximum speed, the kinetic energy must be maximized, which occurs when the potential energy is minimized. Therefore, the maximum speed will occur when the potential energy is at its minimum value.

### Step 4: Identify the Position for Maximum Speed
Since we found that the potential energy is minimized at \( x = 4 \), this is the position where the particle will have its maximum speed.

### Conclusion
Thus, the maximum speed of the particle occurs at:

\[ \text{Maximum speed is at } x = 4 \]

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