The center of mass of a system consisting of two spheres is moving in the positive x direction with a speed of 5.0 m/s. One sphere has a mass of 2.5 kg and the other has a mass of 3.5 kg. The total momentum of the system is :

Options

A. 0
B. `(5.0 kg m//s) hati`
C. `(12.5 kg m//s) hati`
D. `(30 kg m//s) hati`

Explanation

To find the total momentum of the system consisting of two spheres, we can follow these steps:

### Step 1: Identify the masses of the spheres
We have two spheres with the following masses:
- Mass of sphere 1 (m1) = 2.5 kg
- Mass of sphere 2 (m2) = 3.5 kg

### Step 2: Calculate the total mass of the system
The total mass (M) of the system can be calculated by adding the masses of the two spheres:
\[ M = m_1 + m_2 \]
Substituting the values:
\[ M = 2.5 \, \text{kg} + 3.5 \, \text{kg} = 6.0 \, \text{kg} \]

### Step 3: Identify the velocity of the center of mass
The center of mass of the system is moving with a velocity (v) of:
\[ v = 5.0 \, \text{m/s} \]
This velocity is in the positive x-direction.

### Step 4: Calculate the total momentum of the system
The total momentum (P) of the system can be calculated using the formula:
\[ P = M \cdot v \]
Substituting the values we found:
\[ P = 6.0 \, \text{kg} \cdot 5.0 \, \text{m/s} \]
\[ P = 30.0 \, \text{kg m/s} \]

### Step 5: Direction of the momentum
Since the center of mass is moving in the positive x-direction, we can represent the momentum vector as:
\[ P = 30.0 \, \text{kg m/s} \, \hat{i} \]

### Final Answer
The total momentum of the system is:
\[ \mathbf{P} = 30.0 \, \text{kg m/s} \, \hat{i} \]

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