A particle starts with a velocity of `2m//s` and moves in a straight line with a retardation of `0.1m//s^(2)`. The time that it takes to describe `15m` is

Options

A. 10 s
B. 20 s
C. 30 s
D. 40 s

Explanation

To solve the problem step by step, we will use the equations of motion. Here’s how we can approach it:

### Step 1: Identify the given values
- Initial velocity (u) = 2 m/s
- Retardation (a) = -0.1 m/s² (negative because it is retardation)
- Displacement (s) = 15 m

### Step 2: Use the equation of motion
We will use the equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]

Substituting the known values into the equation:
\[ 15 = 2t + \frac{1}{2} (-0.1) t^2 \]

### Step 3: Simplify the equation
This simplifies to:
\[ 15 = 2t - 0.05t^2 \]

To make calculations easier, we can multiply through by 100 to eliminate the decimal:
\[ 1500 = 200t - 5t^2 \]

Rearranging gives:
\[ 5t^2 - 200t + 1500 = 0 \]

### Step 4: Divide through by 5
To simplify further, divide the entire equation by 5:
\[ t^2 - 40t + 300 = 0 \]

### Step 5: Use the quadratic formula
Now, we can use the quadratic formula to find the roots of the equation:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -40, c = 300 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-40)^2 - 4(1)(300) = 1600 - 1200 = 400 \]

Now substituting into the quadratic formula:
\[ t = \frac{40 \pm \sqrt{400}}{2} \]
\[ t = \frac{40 \pm 20}{2} \]

### Step 6: Calculate the two possible values for t
This gives us two possible solutions:
1. \( t = \frac{60}{2} = 30 \) seconds
2. \( t = \frac{20}{2} = 10 \) seconds

### Step 7: Determine the correct time
Since the question asks for the time taken to describe 15 m, we take the smaller value:
\[ t = 10 \text{ seconds} \]

### Final Answer
The time that it takes to describe 15 m is **10 seconds**.

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