The potential energy of a particle of mass 1 kg moving along x-axis given by `U(x)=[(x^(2))/(2)-x]J`. If total mechanical speed (in m/s):-

Options

A. `sqrt(5)`
B. `sqrt(7)`
C. `sqrt(3)`
D. None

Answer

To solve the problem step-by-step, we can follow these steps:

### Step 1: Write down the expression for potential energy.
The potential energy \( U(x) \) of the particle is given by:
\[ U(x) = \frac{x^2}{2} - x \quad \text{(in joules)} \]

### Step 2: Find the expression for total mechanical energy.
Total mechanical energy \( E \) is the sum of kinetic energy \( K \) and potential energy \( U \):
\[ E = K + U \]

### Step 3: Differentiate the potential energy to find the minimum.
To find the position where potential energy is minimum, we differentiate \( U(x) \) with respect to \( x \):
\[ \frac{dU}{dx} = x - 1 \]

### Step 4: Set the derivative to zero to find critical points.
Setting the derivative equal to zero gives:
\[ x - 1 = 0 \implies x = 1 \]

### Step 5: Calculate the potential energy at \( x = 1 \).
Substituting \( x = 1 \) into the potential energy equation:
\[ U(1) = \frac{1^2}{2} - 1 = \frac{1}{2} - 1 = -\frac{1}{2} \quad \text{(in joules)} \]

### Step 6: Determine the total mechanical energy.
At the point where potential energy is minimum, total mechanical energy is:
\[ E = K + U \implies E = K - \frac{1}{2} \]

### Step 7: Express kinetic energy in terms of total mechanical energy.
The kinetic energy \( K \) can be expressed as:
\[ K = E + \frac{1}{2} \]

### Step 8: Use the kinetic energy formula.
Kinetic energy is also given by:
\[ K = \frac{1}{2} mv^2 \]
Since the mass \( m = 1 \) kg:
\[ K = \frac{1}{2} v^2 \]

### Step 9: Set the two expressions for kinetic energy equal.
Equating the two expressions for kinetic energy:
\[ \frac{1}{2} v^2 = E + \frac{1}{2} \]

### Step 10: Solve for \( v^2 \).
Substituting \( E \) from above, we have:
\[ \frac{1}{2} v^2 = K - \frac{1}{2} + \frac{1}{2} \implies \frac{1}{2} v^2 = K \]
Thus,
\[ v^2 = 2K \]

### Step 11: Substitute \( K \) back to find \( v \).
From the previous steps, we know \( K = E + \frac{1}{2} \). Since \( E \) is the total mechanical energy, we can express \( v \) as:
\[ v = \sqrt{2K} \]

### Step 12: Calculate the maximum speed.
Given that \( K = 5 \) (from the earlier calculations), we can find \( v \):
\[ v = \sqrt{5} \]

### Final Answer:
The total mechanical speed is:
\[ v = \sqrt{5} \text{ m/s} \]

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