Fot the given momentume-time (P-t) graphs find the ratio of force on particles A & B.

Answer

To solve the problem of finding the ratio of forces on particles A and B from the given momentum-time (P-t) graphs, we can follow these steps:

### Step 1: Understand the relationship between force and momentum
The force acting on an object is defined as the rate of change of momentum. Mathematically, this is expressed as:
\[ F = \frac{dp}{dt} \]
where \( F \) is the force, \( p \) is the momentum, and \( t \) is time.

### Step 2: Identify the slopes of the P-t graphs
The slope of the momentum-time graph (P-t graph) represents the force. Therefore, we need to determine the slopes of the lines representing forces A and B on the graph.

### Step 3: Determine the angles of the lines
From the problem, we know that:
- The angle of line A (force A) is \( 30^\circ \).
- The angle of line B (force B) is \( 45^\circ \).

### Step 4: Calculate the slopes using the tangent of the angles
The slope of a line in a graph can be calculated using the tangent of the angle it makes with the horizontal axis:
\[ \text{slope} = \tan(\theta) \]
Thus, we can write:
- Slope of line A:
\[ \text{slope}_A = \tan(30^\circ) \]
- Slope of line B:
\[ \text{slope}_B = \tan(45^\circ) \]

### Step 5: Find the values of the tangents
Using known values:
- \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \)
- \( \tan(45^\circ) = 1 \)

### Step 6: Set up the ratio of forces
Since the force is proportional to the slope, we can express the ratio of forces \( F_A \) and \( F_B \) as:
\[ \frac{F_A}{F_B} = \frac{\tan(30^\circ)}{\tan(45^\circ)} \]

### Step 7: Substitute the values of the tangents
Substituting the values we found:
\[ \frac{F_A}{F_B} = \frac{\frac{1}{\sqrt{3}}}{1} = \frac{1}{\sqrt{3}} \]

### Step 8: Express the ratio in a simpler form
This can also be expressed as:
\[ F_A : F_B = 1 : \sqrt{3} \]

### Final Answer
Thus, the ratio of the forces on particles A and B is:
\[ F_A : F_B = 1 : \sqrt{3} \]

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