These questions consists of two statements each printed as Assertion and Reason. While answering these question you are required to choose any one of the following five reponses
(a) If both Assertion and Reason arecorrect and Reason is the correct explanation of Asserrtion.
(b) If both Assertion and Reason are correct but Reason is not the correct explanation of Assertion.
(c ) If Assertion is true but Reason is false.
(d) If Assertion is false but Reason is true.
Assertion A block of mass m starts moving on a rough horizontal surface with a velocity v. It stops due to friction between the block and the surface after moving through a certain distance.
The surface is now tilted to an angle of `30^(@)` with the horizontal and the same block is made to go up on the surface with the same initial velocity v. The decrease in the mechanical energy in the second situation is smaller than that in the first situation.
Reason The coefficient of friction between the block and the surface decreases with the increase in the anglle of inclination.

Explanation

To solve the problem, we need to analyze both the assertion and the reason provided in the question step by step.

### Step 1: Analyze the Assertion
The assertion states that a block of mass \( m \) starts moving on a rough horizontal surface with an initial velocity \( v \) and stops due to friction after moving a certain distance. Then, when the surface is tilted at an angle of \( 30^\circ \), the same block moves up the incline with the same initial velocity \( v \). The assertion claims that the decrease in mechanical energy in this second situation is smaller than in the first situation.

**Explanation:**
- When the block moves on a horizontal surface, it loses mechanical energy due to work done against friction until it stops. The work done against friction can be calculated using the formula:
\[ W = f \cdot d \]
where \( f \) is the frictional force and \( d \) is the distance moved.

- On the inclined plane, the block not only has to overcome friction but also the component of gravitational force acting down the incline. Thus, the energy loss will be due to both friction and gravitational potential energy.

### Step 2: Analyze the Reason
The reason states that the coefficient of friction between the block and the surface decreases with the increase in the angle of inclination.

**Explanation:**
- The coefficient of friction is a property that depends on the materials in contact and does not inherently change with the angle of inclination. It remains constant unless the nature of the surfaces changes. Therefore, the reason is incorrect.

### Step 3: Conclusion
Now, we can conclude the relationship between the assertion and the reason:
- The assertion is **true** because the decrease in mechanical energy when moving up the incline is indeed smaller than when moving on a horizontal surface due to the additional gravitational potential energy component.
- The reason is **false** because the coefficient of friction does not decrease with the angle of inclination.

### Final Answer
Based on the analysis:
- The correct response is **(c)**: If Assertion is true but Reason is false.

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