What is the dimension of new quantity named `''CHIGS''` if it depends only on velocity of an object and mass of object.
(i) CHIGS is proportional to mass of object keeping velocity constant.
(ii) CHIGS is proportional to kinetic energy of object keeping mass constant.

Options

A. Question can be solved by information I only.
B. Question can be solved by information II only
C. Question can be solved by information I and II in combined form only.
D. Question can not be solved by both the informations only.

Answer

To find the dimension of the new quantity named "CHIGS," we will analyze the given information step by step.

### Step 1: Understanding the Proportional Relationships

1. **Given Information:**
- CHIGS is proportional to the mass of the object (m) when velocity (v) is constant.
- CHIGS is proportional to the kinetic energy of the object (KE) when mass is constant.

### Step 2: Analyzing the First Relationship

2. **From the first relationship:**
- If CHIGS (let's denote it as \( C \)) is proportional to mass \( m \) when velocity is constant, we can express this as:
\[ C \propto m \]
- This implies that:
\[ C = k_1 \cdot m \]
where \( k_1 \) is a proportionality constant.

### Step 3: Analyzing the Second Relationship

3. **From the second relationship:**
- Kinetic energy (KE) is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
- If CHIGS is proportional to kinetic energy when mass is constant, we can express this as:
\[ C \propto KE \]
- Therefore, we can write:
\[ C = k_2 \cdot KE \]
where \( k_2 \) is another proportionality constant.

### Step 4: Combining the Relationships

4. **Combining the two relationships:**
- From the first relationship, we have \( C = k_1 \cdot m \).
- From the second relationship, substituting for kinetic energy, we have:
\[ C = k_2 \cdot \left(\frac{1}{2} mv^2\right) \]
- Setting these equal gives:
\[ k_1 \cdot m = k_2 \cdot \left(\frac{1}{2} mv^2\right) \]

### Step 5: Simplifying the Equation

5. **Simplifying the equation:**
- We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ k_1 = k_2 \cdot \frac{1}{2} v^2 \]
- This shows that \( C \) depends on both mass and velocity.

### Step 6: Finding the Dimensions

6. **Finding the dimensions of CHIGS:**
- The dimension of mass \( [m] \) is \( M \).
- The dimension of velocity \( [v] \) is \( LT^{-1} \).
- The kinetic energy \( [KE] \) has dimensions of \( [M][L^2][T^{-2}] \).
- Therefore, we can express CHIGS in terms of its dimensions:
\[ C \propto m \cdot v^2 \]
- Thus, the dimensions of CHIGS can be expressed as:
\[ [C] = [M] \cdot [L^2][T^{-2}] = M \cdot (LT^{-1})^2 = M \cdot L^2 T^{-2} \]

### Final Answer

The dimension of the quantity CHIGS is:
\[ [M^1 L^2 T^{-2}] \]