A particle with total energy E is moving in a potential energy region U(x). Motion of the particle is restricted to the region when

Options

A. `U(x) gt E`
B. `E(x) lt E`
C. `U(x) =0`
D. `U(x) leE`

Answer

To solve the problem, we need to analyze the motion of a particle with total energy \( E \) in a potential energy region \( U(x) \). The key concept here is that the total energy of the particle is the sum of its kinetic energy and potential energy.

### Step-by-Step Solution:

1. **Understanding Total Energy**:
The total energy \( E \) of the particle is given by the equation:
\[ E = K + U(x) \]
where \( K \) is the kinetic energy and \( U(x) \) is the potential energy at position \( x \).

2. **Expressing Kinetic Energy**:
Rearranging the equation gives us:
\[ K = E - U(x) \]
This equation tells us that the kinetic energy \( K \) is equal to the total energy \( E \) minus the potential energy \( U(x) \).

3. **Condition for Motion**:
Kinetic energy must always be non-negative, as it cannot be less than zero:
\[ K \geq 0 \]
Substituting the expression for kinetic energy, we have:
\[ E - U(x) \geq 0 \]

4. **Rearranging the Inequality**:
From the inequality \( E - U(x) \geq 0 \), we can rearrange it to find:
\[ E \geq U(x) \]

5. **Conclusion**:
This means that the motion of the particle is restricted to the region where the total energy \( E \) is greater than or equal to the potential energy \( U(x) \). Therefore, the particle can only exist in regions where its total energy is greater than or equal to the potential energy at that point.

### Final Answer:
The motion of the particle is restricted to the region where:
\[ E \geq U(x) \]