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 relationship between the total energy \( E \), potential energy \( U(x) \), and kinetic energy \( K(x) \) of a particle.

### Step-by-Step Solution:

1. **Understanding Total Energy**:
The total energy \( E \) of a particle is given by the sum of its kinetic energy \( K \) and potential energy \( U \):
\[ E = K + U \]

2. **Expressing Kinetic Energy**:
Rearranging the equation, we can express the kinetic energy in terms of total energy and potential energy:
\[ K = E - U(x) \]

3. **Condition for Motion**:
For the particle to be in motion, its kinetic energy must be greater than zero:
\[ K > 0 \implies E - U(x) > 0 \]

4. **Inequality Derivation**:
From the inequality \( E - U(x) > 0 \), we can derive:
\[ E > U(x) \]

5. **Conclusion**:
This means that the motion of the particle is restricted to the region where the potential energy \( U(x) \) is less than the total energy \( E \):
\[ U(x) < E \]

Thus, the motion of the particle is restricted to the region where \( U(x) \) is less than \( E \).

### Final Answer:
The motion of the particle is restricted to the region when \( U(x) < E \).