Consider a one-dimensional motion of a particle with total energy E. There are four regions A, B, C and D is which the relation between potential energy U, kinetic energy (K) and total energy E is as given below
RegionA:`UgtE` Region B:`UltE`
Region C:`KltE` Region D:`UgtE`
State with reason in each case whether a particle can be found in the given region or not.

Options

A. Region A
B. Region B
C. Region C
D. Region D

Explanation

To analyze whether a particle can be found in each of the given regions (A, B, C, and D) based on the relationships between potential energy (U), kinetic energy (K), and total energy (E), we can follow these steps:

### Step-by-Step Solution:

1. **Region A: \( U > E \)**
- In this region, the potential energy is greater than the total energy.
- The total energy \( E \) is the sum of kinetic energy \( K \) and potential energy \( U \):
\[ E = K + U \]
- Rearranging gives:
\[ K = E - U \]
- Since \( U > E \), this implies \( K = E - U < 0 \).
- Kinetic energy cannot be negative, hence a particle **cannot be found in Region A**.

2. **Region B: \( U < E \)**
- Here, the potential energy is less than the total energy.
- Using the same equation for total energy:
\[ K = E - U \]
- Since \( U < E \), it follows that \( K = E - U > 0 \).
- Kinetic energy is positive, therefore a particle **can be found in Region B**.

3. **Region C: \( K < E \)**
- In this region, kinetic energy is less than total energy.
- Again, using the total energy equation:
\[ K = E - U \]
- If \( K < E \), then \( E - U > K \) implies \( U < E \).
- Since potential energy can be positive and \( K \) is positive, a particle **can be found in Region C**.

4. **Region D: \( U > E \)**
- Similar to Region A, potential energy is greater than total energy.
- This means:
\[ K = E - U < 0 \]
- Again, kinetic energy cannot be negative, hence a particle **cannot be found in Region D**.

### Summary of Results:
- **Region A**: Particle cannot be found (K < 0).
- **Region B**: Particle can be found (K > 0).
- **Region C**: Particle can be found (K > 0).
- **Region D**: Particle cannot be found (K < 0).