The raindrops are hitting the back of a man walking at a speed of `5km//hr`. If he now starts running in the same direction with a constant acceleration, the magnitude of the velocity of the rain with respect to him will

Options

A. gradually increase
B. gradually decrease
C. first decrease then increase
D. first increase then decrease

Explanation

To solve the problem, we need to analyze the situation of the man walking and then running in relation to the rain. Let's break it down step by step:

### Step 1: Understand the Initial Situation
The man is initially walking at a speed of 5 km/hr. The rain is falling vertically downwards, and since the raindrops are hitting the back of the man, we can infer that the rain has a horizontal component of velocity equal to the speed of the man (5 km/hr) in the opposite direction.

### Step 2: Define the Velocity of the Rain
Let’s denote:
- \( V_m = 5 \) km/hr (velocity of the man)
- \( V_r \) = velocity of the rain (which has both vertical and horizontal components)

Since the rain is hitting the man from behind, we can say that the horizontal component of the rain's velocity must equal the man's speed in the opposite direction. Therefore, we can assume:
- The horizontal component of the rain's velocity \( V_{rh} = -5 \) km/hr (negative because it’s in the opposite direction to the man’s motion).

### Step 3: The Man Starts Running
Now, the man starts running in the same direction with a constant acceleration. Let’s denote his running speed as \( V_{mr} \). As he accelerates, his speed will increase beyond 5 km/hr.

### Step 4: Relative Velocity of Rain with Respect to the Man
To find the relative velocity of the rain with respect to the man, we use the formula:
\[ V_{rm} = V_r - V_m \]
Where:
- \( V_{rm} \) is the velocity of the rain with respect to the man.
- \( V_r \) is the velocity of the rain (which has a horizontal component of -5 km/hr).
- \( V_m \) is the velocity of the man.

As the man accelerates, his speed \( V_m \) increases. Therefore, the relative velocity can be expressed as:
\[ V_{rm} = V_{rh} - V_{mr} \]
Since \( V_{rh} = -5 \) km/hr, we have:
\[ V_{rm} = -5 - V_{mr} \]

### Step 5: Analyze the Change in Relative Velocity
As the man runs faster (increasing \( V_{mr} \)), the term \( -V_{mr} \) becomes more negative, which means \( V_{rm} \) will decrease in magnitude.

Initially, when the man is walking at 5 km/hr, the rain appears to be falling at a speed of 5 km/hr relative to him. As he accelerates, the relative velocity \( V_{rm} \) will decrease because he is moving away from the rain's horizontal component.

### Step 6: Conclusion
Thus, the magnitude of the velocity of the rain with respect to the man will initially be 5 km/hr and will decrease as he accelerates. Therefore, the correct option is that the relative velocity of the rain with respect to the man will gradually decrease.

### Final Answer
The magnitude of the velocity of the rain with respect to him will gradually decrease.

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