A block of mass `10 kg` is moving in x-direction with a constant speed of `10 m//s`. it is subjected to a retarding force `F=-0.1 x J//m`. During its travel from `x =20m` to `x =30 m`. Its final kinetic energy will be .

Options

A. 250 J
B. 275 J
C. 450 J
D. 475 J

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the given values
- Mass of the block, \( m = 10 \, \text{kg} \)
- Initial speed of the block, \( v = 10 \, \text{m/s} \)
- Retarding force, \( F = -0.1x \, \text{J/m} \)
- Initial position, \( x_i = 20 \, \text{m} \)
- Final position, \( x_f = 30 \, \text{m} \)

### Step 2: Calculate the initial kinetic energy
The initial kinetic energy (\( KE_i \)) can be calculated using the formula:
\[ KE_i = \frac{1}{2} mv^2 \]
Substituting the values:
\[ KE_i = \frac{1}{2} \times 10 \, \text{kg} \times (10 \, \text{m/s})^2 = \frac{1}{2} \times 10 \times 100 = 500 \, \text{J} \]

### Step 3: Calculate the work done by the retarding force
The work done (\( W \)) by the force as the block moves from \( x = 20 \, \text{m} \) to \( x = 30 \, \text{m} \) can be calculated using the integral of the force over the distance:
\[ W = \int_{x_i}^{x_f} F \, dx \]
Substituting the expression for \( F \):
\[ W = \int_{20}^{30} (-0.1x) \, dx \]
Calculating the integral:
\[ W = -0.1 \int_{20}^{30} x \, dx = -0.1 \left[ \frac{x^2}{2} \right]_{20}^{30} \]
Calculating the limits:
\[ = -0.1 \left( \frac{30^2}{2} - \frac{20^2}{2} \right) = -0.1 \left( \frac{900}{2} - \frac{400}{2} \right) = -0.1 \left( 450 - 200 \right) = -0.1 \times 250 = -25 \, \text{J} \]

### Step 4: Calculate the final kinetic energy
Using the work-energy principle, the final kinetic energy (\( KE_f \)) can be calculated as:
\[ KE_f = KE_i + W \]
Substituting the values:
\[ KE_f = 500 \, \text{J} + (-25 \, \text{J}) = 500 \, \text{J} - 25 \, \text{J} = 475 \, \text{J} \]

### Final Answer
The final kinetic energy of the block when it travels from \( x = 20 \, \text{m} \) to \( x = 30 \, \text{m} \) is \( 475 \, \text{J} \).

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