A particle with total energy `E` moves in one direction in a region where, the potential energy is `U` The acceleration of the particle is zero, where,

Options

A. `U=E`
B. `U=0`
C. `(dU)/(dx)=0`
D. `(d^(2)U)/(dx^(2))=0`

Explanation

To solve the problem, we need to analyze the conditions under which the acceleration of a particle is zero. Here’s the step-by-step solution:

### Step 1: Understand the relationship between force and potential energy
The force acting on a particle in a potential energy field is given by the negative gradient of the potential energy. Mathematically, this is expressed as:
\[ F = -\frac{dU}{dx} \]
where \( U \) is the potential energy and \( x \) is the position.

### Step 2: Set the condition for zero acceleration
For a particle to have zero acceleration, the net force acting on it must also be zero. Therefore, we set the force equation to zero:
\[ F = 0 \]

### Step 3: Substitute the force equation
From the force equation, we have:
\[ -\frac{dU}{dx} = 0 \]

### Step 4: Solve for the potential energy gradient
To satisfy the equation, we need:
\[ \frac{dU}{dx} = 0 \]
This indicates that the potential energy \( U \) is constant with respect to position \( x \).

### Step 5: Conclusion
Thus, the acceleration of the particle is zero when the derivative of the potential energy with respect to position is zero. This means that in the region where the particle is moving, the potential energy does not change with position.

### Final Answer:
The acceleration of the particle is zero where:
\[ \frac{dU}{dx} = 0 \]

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