A body of mass 5 kg starts from the origin with an initial velocity `bar(u)=(30hati+40hatj)ms^(-1)` .If a constant force `(-6hati-5hatj)N` acts on the body, the time in velocity, which the y-component of the velocity becomes zero is.

Options

A. 1. 5s
B. 2. 20s
C. 3. 40s
D. 5. 80s

Explanation

To solve the problem, we need to find the time at which the y-component of the velocity of the body becomes zero. Here’s how we can approach the solution step by step:

### Step 1: Identify the given data
- Mass of the body, \( m = 5 \, \text{kg} \)
- Initial velocity, \( \mathbf{u} = 30 \hat{i} + 40 \hat{j} \, \text{m/s} \)
- Constant force acting on the body, \( \mathbf{F} = -6 \hat{i} - 5 \hat{j} \, \text{N} \)

### Step 2: Calculate the acceleration
Using Newton's second law, the acceleration \( \mathbf{a} \) can be calculated using the formula:
\[ \mathbf{F} = m \mathbf{a} \]
Rearranging gives:
\[ \mathbf{a} = \frac{\mathbf{F}}{m} \]
Substituting the values:
\[ \mathbf{a} = \frac{-6 \hat{i} - 5 \hat{j}}{5} = -1.2 \hat{i} - 1 \hat{j} \, \text{m/s}^2 \]
Thus, the acceleration in the y-direction is:
\[ a_y = -1 \, \text{m/s}^2 \]

### Step 3: Write the equation for the y-component of velocity
The y-component of the velocity \( v_y \) can be expressed as:
\[ v_y = u_y + a_y \cdot t \]
Where:
- \( u_y = 40 \, \text{m/s} \) (initial y-component of velocity)
- \( a_y = -1 \, \text{m/s}^2 \)

### Step 4: Set the y-component of velocity to zero
We want to find the time \( t \) when \( v_y = 0 \):
\[ 0 = 40 - 1 \cdot t \]

### Step 5: Solve for time \( t \)
Rearranging the equation gives:
\[ t = 40 \, \text{s} \]

### Conclusion
The time at which the y-component of the velocity becomes zero is \( t = 40 \, \text{s} \).

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