A block of mass m moving at a speed v compresses a spring throgh a distance x before its speed is halved. Find the spring constant of the spring.

Explanation

To solve the problem where a block of mass \( m \) moving at a speed \( v \) compresses a spring through a distance \( x \) before its speed is halved, we will use the principle of conservation of energy. Here’s a step-by-step solution:

### Step 1: Identify the initial kinetic energy of the block
The initial kinetic energy (\( KE_i \)) of the block can be calculated using the formula:
\[ KE_i = \frac{1}{2} m v^2 \]

### Step 2: Identify the final kinetic energy of the block
When the block compresses the spring and its speed is halved, its final speed becomes \( \frac{v}{2} \). The final kinetic energy (\( KE_f \)) is:
\[ KE_f = \frac{1}{2} m \left(\frac{v}{2}\right)^2 = \frac{1}{2} m \frac{v^2}{4} = \frac{1}{8} m v^2 \]

### Step 3: Calculate the loss in kinetic energy
The loss in kinetic energy (\( \Delta KE \)) as the block compresses the spring is given by:
\[ \Delta KE = KE_i - KE_f = \frac{1}{2} m v^2 - \frac{1}{8} m v^2 \]
To simplify this, we need a common denominator:
\[ \Delta KE = \frac{4}{8} m v^2 - \frac{1}{8} m v^2 = \frac{3}{8} m v^2 \]

### Step 4: Calculate the potential energy stored in the spring
The potential energy (\( PE \)) stored in the spring when it is compressed by a distance \( x \) is given by:
\[ PE = \frac{1}{2} k x^2 \]

### Step 5: Apply the conservation of energy principle
According to the conservation of energy, the loss in kinetic energy is equal to the gain in potential energy:
\[ \Delta KE = PE \]
Substituting the expressions we found:
\[ \frac{3}{8} m v^2 = \frac{1}{2} k x^2 \]

### Step 6: Solve for the spring constant \( k \)
Rearranging the equation to solve for \( k \):
\[ k x^2 = \frac{3}{4} m v^2 \]
\[ k = \frac{3 m v^2}{4 x^2} \]

### Final Answer
The spring constant \( k \) is given by:
\[ k = \frac{3 m v^2}{4 x^2} \]

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