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Step-by-Step Solution
Step 1: Identify the Known Quantities
• Mass of the block, $m = 2\,\text{kg}$
• Initial speed of the block, $v = 4\,\text{m/s}$
• Kinetic friction force, $f = 15\,\text{N}$
• Spring constant, $k = 10{,}000\,\text{N/m}$
• Let the compression in the spring be $x$.
Step 2: Express Energy Conservation
As the block moves and compresses the spring, its initial kinetic energy is completely used up in two ways:
1) The work done against friction, which is $f \cdot x$.
2) The potential energy stored in the compressed spring, which is $ \frac{1}{2} k x^2 $.
Step 3: Write the Energy Balance Equation
Initial kinetic energy of the block is
$ \frac{1}{2} m v^2 = \frac{1}{2} \times 2 \times (4)^2 = 16\,\text{J}. $
Energy conservation gives:
$ \frac{1}{2} m v^2 = \frac{1}{2} k x^2 + f \cdot x. $
Substituting the values:
$ 16 = \frac{1}{2} \times 10{,}000 \times x^2 + 15\,x. $
Simplifying further:
$ 16 = 5{,}000\,x^2 + 15\,x. $
Rearranging to form a quadratic equation:
$ 5{,}000\,x^2 + 15\,x - 16 = 0. $
Step 4: Solve the Quadratic Equation
The quadratic equation $ 5{,}000\,x^2 + 15\,x - 16 = 0 $ can be solved using the quadratic formula:
$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $
where $ a = 5{,}000, \; b = 15, \; c = -16. $
Substituting these values:
$ x = \frac{-15 \pm \sqrt{(15)^2 - 4 \times 5{,}000 \times (-16)}}{2 \times 5{,}000}. $
The physically meaningful (positive) root is:
$ x = 0.055\,\text{m}. $
Step 5: Convert the Result to Centimeters
$ 0.055\,\text{m} = 0.055 \times 100\,\text{cm} = 5.5\,\text{cm}. $
Final Answer
The spring compresses by $5.5\,\text{cm}.$
