A body of mass 5 kg starts from the origin with an initial velocity `vec(u) = 30 hat(i) + 40hat(j) ms^(-1)`. If a constant force `vec(F) = - (hat(i) + 5hat(j))N` acts on the baby , the time in which the y-component of the velocity becomes zero it .

Options

A. 5 seconds
B. 20 seconds
C. 40 seconds
D. 80 seconds

Explanation

To solve the problem step by step, we will follow the physics principles involving motion and forces.

### Step 1: Identify the initial conditions
The body has a mass \( m = 5 \, \text{kg} \) and an initial velocity given by:
\[ \vec{u} = 30 \hat{i} + 40 \hat{j} \, \text{m/s} \]
From this, we can extract the initial velocity components:
- \( u_x = 30 \, \text{m/s} \) (x-component)
- \( u_y = 40 \, \text{m/s} \) (y-component)

### Step 2: Identify the force acting on the body
The constant force acting on the body is given by:
\[ \vec{F} = -\hat{i} + 5\hat{j} \, \text{N} \]
From this, we can find the components of the force:
- \( F_x = -1 \, \text{N} \) (x-component)
- \( F_y = 5 \, \text{N} \) (y-component)

### Step 3: Calculate the acceleration using Newton's second law
According to Newton's second law, \( \vec{F} = m \vec{a} \), we can find the acceleration components:
- For the y-direction:
\[ F_y = m a_y \implies 5 = 5 a_y \implies a_y = \frac{5}{5} = 1 \, \text{m/s}^2 \]
However, since the force in the y-direction is acting upwards, we need to consider the direction of the force:
\[ F_y = -5 \implies 5 = 5 a_y \implies a_y = -1 \, \text{m/s}^2 \]
Thus, the acceleration in the y-direction is \( a_y = -1 \, \text{m/s}^2 \).

### Step 4: Use the first equation of motion to find the time when the y-component of velocity becomes zero
We will use the first equation of motion:
\[ v_y = u_y + a_y t \]
We want to find the time \( t \) when \( v_y = 0 \):
\[ 0 = 40 + (-1)t \]
Rearranging gives:
\[ t = 40 \, \text{s} \]

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

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