(A) : The apparent weight of a body in an elevator moving with some downward acceleration is always less than the actual weight of body.
(R) : Normal reaction force by the floor of elevator on the body is less than the weight of the body when elevator is moving down

Options

A. Both (A) and (R) are true and (R) is the correct explanation of (A)
B. Both (A) and (R) are true and (R) is not the correct explanation of A
C. (A) is true but ( R) is false
D. Both (A) and (R ) are false

Answer

To solve the given question, we need to analyze both the assertion (A) and the reason (R) provided in the context of an elevator moving downward with acceleration.

### Step-by-Step Solution:

1. **Understanding the Assertion (A)**:
- The assertion states that "The apparent weight of a body in an elevator moving with some downward acceleration is always less than the actual weight of the body."
- The apparent weight is the weight that a person feels, which can change based on the motion of the elevator.

2. **Understanding the Reason (R)**:
- The reason states that "Normal reaction force by the floor of the elevator on the body is less than the weight of the body when the elevator is moving down."
- The normal force is the force exerted by a surface to support the weight of an object resting on it.

3. **Analyzing Forces in the Elevator**:
- Let \( m \) be the mass of the body, \( g \) be the acceleration due to gravity, and \( a \) be the downward acceleration of the elevator.
- The actual weight of the body is given by \( W = mg \).
- When the elevator is moving downward with acceleration \( a \), the forces acting on the body are:
- Downward force due to gravity: \( mg \)
- Upward normal force: \( N \)

4. **Applying Newton's Second Law**:
- According to Newton's second law, the net force acting on the body can be expressed as:
\[ mg - N = ma \]
- Rearranging this equation gives:
\[ N = mg - ma \]
- This shows that the normal force \( N \) (apparent weight) is less than the actual weight \( mg \) because \( ma \) is a positive quantity.

5. **Conclusion**:
- From the equation \( N = mg - ma \), we can conclude that the normal force \( N \) (apparent weight) is indeed less than the actual weight \( mg \) when the elevator is moving downward with acceleration \( a \).
- Therefore, both the assertion (A) and the reason (R) are true, and the reason correctly explains the assertion.

### Final Answer:
Both (A) and (R) are true, and (R) is the correct explanation of (A).

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