A block of mass m is initially moving to the right on a horizontal frictionless surface at a speed v. It then compresses a spring of spring constant k. At the instant when the kinetic energy of the block is equal to the potential energy of the spring, the spring is compressed a distance of :

Options

A. (a)`vsqrt(m //2k)`
B. (b)`vsqrt(m //k)`
C. (c)`vsqrt(m //4k)`
D. (d)`mv^2//4k`

Explanation

To solve the problem step by step, we will use the principle of conservation of energy.

### Step 1: Understand the Initial Conditions
The block of mass \( m \) is moving with an initial speed \( v \) on a frictionless surface. As it compresses the spring, it loses kinetic energy and gains potential energy.

### Step 2: Write the Expression for Kinetic Energy
The initial kinetic energy (KE) of the block is given by:
\[ KE = \frac{1}{2} mv^2 \]

### Step 3: Write the Expression for Potential Energy
The potential energy (PE) stored in the spring when compressed by a distance \( x \) is given by:
\[ PE = \frac{1}{2} k x^2 \]

### Step 4: Set Kinetic Energy Equal to Potential Energy
According to the problem, at the instant when the kinetic energy of the block is equal to the potential energy of the spring, we can write:
\[ \frac{1}{2} mv^2 = \frac{1}{2} k x^2 \]

### Step 5: Simplify the Equation
We can eliminate the \( \frac{1}{2} \) from both sides:
\[ mv^2 = k x^2 \]

### Step 6: Solve for \( x \)
Rearranging the equation to solve for \( x \):
\[ x^2 = \frac{mv^2}{k} \]
Taking the square root of both sides gives:
\[ x = \sqrt{\frac{mv^2}{k}} \]

### Step 7: Final Expression for Compression
Thus, the distance \( x \) at which the spring is compressed when the kinetic energy equals the potential energy is:
\[ x = v \sqrt{\frac{m}{k}} \]

### Conclusion
The distance the spring is compressed when the kinetic energy of the block equals the potential energy of the spring is:
\[ x = v \sqrt{\frac{m}{k}} \]