To solve this problem, we need to analyze the motion of both the monkey and the platform using the principles of conservation of momentum and relative motion. Let's break down each part of the question step by step.

Understanding the System

We have a monkey with a mass of 3 kg sitting on a platform that has a mass of 6 kg. When the monkey moves, it will exert a force on the platform, causing both to move. The key concepts we will use here are:

• Conservation of momentum
• Relative motion

Analyzing the Ratios

Let's denote:

• m_m = mass of the monkey = 3 kg
• m_p = mass of the platform = 6 kg

When the monkey moves to one side, the platform will move in the opposite direction due to the conservation of momentum. The total momentum before and after the monkey moves must remain constant.

1. Ratio of Distance Travelled by Monkey to Distance Travelled by Platform (P)

According to the conservation of momentum:

m_m * v_m = m_p * v_p

Where v_m is the speed of the monkey and v_p is the speed of the platform. The distances travelled by the monkey (d_m) and the platform (d_p) are related to their speeds:

d_m / d_p = v_m / v_p

From the momentum equation, we can derive:

v_m / v_p = m_p / m_m = 6 / 3 = 2

This means the monkey travels twice the distance compared to the platform:

Ratio (P): 2 : 1

2. Ratio of Speed of Platform to Speed of Monkey (Q)

From our previous calculation, we found:

v_p / v_m = m_m / m_p = 3 / 6 = 1 / 2

This indicates that the speed of the platform is half that of the monkey:

Ratio (Q): 1 : 2

3. Ratio of Distance Travelled by Monkey with Respect to Platform and Distance Travelled by Platform (R)

When considering the distance travelled by the monkey relative to the platform, we can express it as:

d_m (relative) = d_m - d_p

Since we established that d_m = 2 * d_p, we can substitute:

d_m (relative) = 2 * d_p - d_p = d_p

This means the distance travelled by the monkey with respect to the platform is equal to the distance travelled by the platform:

Ratio (R): 1 : 1

4. Ratio of Speed of Platform and Speed of Monkey with Respect to Platform (S)

When we consider the speeds of both the monkey and the platform relative to the platform, the speed of the platform is 0 (since it's the reference point), and the speed of the monkey is v_m:

Ratio (S): 0 : v_m

Final Matching

Now, let's match the ratios with the options provided:

• P = 2 : 1 (matches with 2)
• Q = 1 : 2 (matches with 4)
• R = 1 : 1 (not listed, but we derived it)
• S = 0 : v_m (not listed, but we derived it)

Based on the analysis, the correct matching is:

Codes: (C) 2 4 1 3