Five persons A, B, C, D & E are pulling a cart of mass 100kg on a smooth surface and cart is moving with acceleration `3m//s^2` in east direction. When person A' stops pulling, it moves with acceleration `1m//s^2`in the west direction. When person 'B' stops pulling, it moves with accele-ration `24m//s^2` in the north direction. The magnitude of acceleration of the cart when only A & B pull the cart keeping their directions same as the old direction is

Options

A. `26 m//s^(2)`
B. `3sqrt(71) m//s^(2)`
C. `25 m//s^(2)`
D. `30 m//s^(2)`

Answer

To solve the problem step by step, we will analyze the forces acting on the cart when different persons are pulling it and derive the required acceleration when only A and B are pulling.

### Step 1: Understand the initial conditions
The cart has a mass of 100 kg and is initially being pulled by five persons (A, B, C, D, and E) with an acceleration of 3 m/s² towards the east.

### Step 2: Define the directions
We will assign unit vectors to the directions:
- East: \( \hat{i} \)
- West: \( -\hat{i} \)
- North: \( \hat{j} \)
- South: \( -\hat{j} \)

### Step 3: Analyze the situation when A stops pulling
When person A stops pulling, the cart accelerates at 1 m/s² towards the west. This means the net acceleration due to the pulling of A is:
\[ A_A = 3 \hat{i} + A_A = -1 \hat{i} \]
From this, we can deduce:
\[ A_A = 4 \hat{i} \]

### Step 4: Analyze the situation when B stops pulling
When person B stops pulling, the cart accelerates at 24 m/s² towards the north. This means the net acceleration due to the pulling of B is:
\[ A_B = 3 \hat{i} + A_B = 24 \hat{j} \]
From this, we can deduce:
\[ A_B = 3 \hat{i} - 24 \hat{j} \]

### Step 5: Calculate the contributions of A and B
Now, we can find the total acceleration when both A and B are pulling:
\[ A_{total} = A_A + A_B \]
Substituting the values we found:
\[ A_{total} = (4 \hat{i}) + (3 \hat{i} - 24 \hat{j}) \]
\[ A_{total} = (4 + 3) \hat{i} - 24 \hat{j} \]
\[ A_{total} = 7 \hat{i} - 24 \hat{j} \]

### Step 6: Calculate the magnitude of the total acceleration
To find the magnitude of the acceleration vector \( A_{total} \):
\[ |A_{total}| = \sqrt{(7)^2 + (-24)^2} \]
\[ |A_{total}| = \sqrt{49 + 576} \]
\[ |A_{total}| = \sqrt{625} \]
\[ |A_{total}| = 25 \, \text{m/s}^2 \]

### Final Answer
The magnitude of the acceleration of the cart when only A and B pull the cart is **25 m/s²**.

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