A particle has an initial velocity of `4hat(i)+4hat(j)`m//s and an acceleration of `-0.4` `hat(i)` m//s^(2) at what time will its speed be 5 m//s?

Options

A. `2.5` sec
B. `17.5` sec
C. `7 sqrt(2)` sec
D. `8.5` sec

Explanation

To solve the problem, we need to find the time at which the speed of the particle becomes 5 m/s given its initial velocity and acceleration.

### Step 1: Understand the initial conditions
The initial velocity of the particle is given as:
\[ \vec{u} = 4\hat{i} + 4\hat{j} \, \text{m/s} \]
This means the initial velocity in the x-direction (\(u_x\)) is 4 m/s and in the y-direction (\(u_y\)) is also 4 m/s.

The acceleration is given as:
\[ \vec{a} = -0.4\hat{i} \, \text{m/s}^2 \]
This indicates that there is a negative acceleration in the x-direction, while the y-direction has no acceleration.

### Step 2: Determine the velocity components
Since there is no acceleration in the y-direction, the velocity in the y-direction will remain constant:
\[ v_y = u_y = 4 \, \text{m/s} \]

For the x-direction, the velocity at time \(t\) can be expressed as:
\[ v_x = u_x + a_x t = 4 - 0.4t \]

### Step 3: Calculate the speed
The speed of the particle is given by the magnitude of the velocity vector:
\[ \text{Speed} = \sqrt{v_x^2 + v_y^2} \]
We want to find the time \(t\) when the speed is 5 m/s:
\[ 5 = \sqrt{(4 - 0.4t)^2 + 4^2} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 25 = (4 - 0.4t)^2 + 16 \]
Subtracting 16 from both sides:
\[ 9 = (4 - 0.4t)^2 \]

### Step 5: Take the square root
Taking the square root of both sides gives two equations:
\[ 4 - 0.4t = 3 \quad \text{or} \quad 4 - 0.4t = -3 \]

### Step 6: Solve for \(t\)
1. For the first equation:
\[ 4 - 0.4t = 3 \implies 0.4t = 4 - 3 = 1 \implies t = \frac{1}{0.4} = 2.5 \, \text{s} \]

2. For the second equation:
\[ 4 - 0.4t = -3 \implies 0.4t = 4 + 3 = 7 \implies t = \frac{7}{0.4} = 17.5 \, \text{s} \]

### Step 7: Conclusion
The times at which the speed of the particle is 5 m/s are:
\[ t = 2.5 \, \text{s} \quad \text{and} \quad t = 17.5 \, \text{s} \]

### Final Answer
The time at which the particle's speed is 5 m/s is either 2.5 seconds or 17.5 seconds.

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