A 2 kg block slides on a horizontal floor with a speed of 4 m/s. It strikes an uncompressed spring, and compresses it till the block is motionless. The kinetic friction force is 15 N and spring constant is 10000 N/m. The spring is compressed by (in cm):

Options

A. (a)2.5
B. (b)`11.0`
C. (c)`8.5`
D. (d)`5.5`

Explanation

To solve the problem step-by-step, we will use the concepts of energy conservation and work done against friction.

### Step 1: Identify the Given Data
- Mass of the block (m) = 2 kg
- Initial speed of the block (v) = 4 m/s
- Kinetic friction force (F_friction) = 15 N
- Spring constant (k) = 10,000 N/m

### Step 2: Calculate the Initial Kinetic Energy (KE)
The kinetic energy of the block can be calculated using the formula:
\[ KE = \frac{1}{2} m v^2 \]
Substituting the values:
\[ KE = \frac{1}{2} \times 2 \, \text{kg} \times (4 \, \text{m/s})^2 = \frac{1}{2} \times 2 \times 16 = 16 \, \text{J} \]

### Step 3: Set Up the Energy Conservation Equation
When the block compresses the spring, the kinetic energy is converted into the potential energy of the spring and the work done against friction. The equation can be set up as:
\[ KE = PE + W \]
Where:
- \(PE = \frac{1}{2} k x^2\) (Potential energy of the spring)
- \(W = F_{friction} \cdot x\) (Work done against friction)

So, we have:
\[ 16 = \frac{1}{2} k x^2 + F_{friction} \cdot x \]

### Step 4: Substitute the Known Values
Substituting the values of \(k\) and \(F_{friction}\):
\[ 16 = \frac{1}{2} \times 10000 \, \text{N/m} \cdot x^2 + 15 \, \text{N} \cdot x \]
This simplifies to:
\[ 16 = 5000 x^2 + 15x \]

### Step 5: Rearrange the Equation
Rearranging the equation gives us a standard quadratic form:
\[ 5000 x^2 + 15x - 16 = 0 \]

### Step 6: Solve the Quadratic Equation
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 5000\), \(b = 15\), and \(c = -16\):
1. Calculate the discriminant:
\[ D = b^2 - 4ac = 15^2 - 4 \times 5000 \times (-16) = 225 + 320000 = 320225 \]
2. Calculate \(x\):
\[ x = \frac{-15 \pm \sqrt{320225}}{2 \times 5000} \]
\[ x = \frac{-15 \pm 566.3}{10000} \]
Taking the positive root:
\[ x = \frac{551.3}{10000} = 0.05513 \, \text{m} \]

### Step 7: Convert to Centimeters
To convert meters to centimeters:
\[ x = 0.05513 \, \text{m} \times 100 = 5.513 \, \text{cm} \]

### Final Answer
The spring is compressed by approximately **5.5 cm**.

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