A mass of `0.5 kg` moving with a speed of `1.5 m//s` on a horizontal smooth surface, collides with a nearly weightless spring of force constant `k =50 N//m` The maximum compression of the spring would be.

Options

A. a)`0.15 m`
B. b)`0.12m`
C. c)`0.5 m`
D. d)`0.25 m`

Explanation

To solve the problem of finding the maximum compression of the spring when a mass collides with it, we can follow these steps:

### Step 1: Identify the given values
- Mass (m) = 0.5 kg
- Initial speed (v) = 1.5 m/s
- Spring constant (k) = 50 N/m

### Step 2: Write the formula for kinetic energy (KE)
The kinetic energy of the mass before it collides with the spring is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]

### Step 3: Calculate the initial kinetic energy
Substituting the values into the kinetic energy formula:
\[ KE = \frac{1}{2} \times 0.5 \, \text{kg} \times (1.5 \, \text{m/s})^2 \]
\[ KE = \frac{1}{2} \times 0.5 \times 2.25 = 0.5625 \, \text{J} \]

### Step 4: Write the formula for potential energy (PE) stored in the spring
The potential energy stored in the spring at maximum compression (x) is given by:
\[ PE = \frac{1}{2} kx^2 \]

### Step 5: Set the kinetic energy equal to the potential energy
At maximum compression, all the kinetic energy will be converted into potential energy:
\[ \frac{1}{2} mv^2 = \frac{1}{2} kx^2 \]
We can cancel the \(\frac{1}{2}\) from both sides:
\[ mv^2 = kx^2 \]

### Step 6: Rearrange the equation to solve for x
\[ x^2 = \frac{mv^2}{k} \]
\[ x = \sqrt{\frac{mv^2}{k}} \]

### Step 7: Substitute the known values into the equation
\[ x = \sqrt{\frac{0.5 \, \text{kg} \times (1.5 \, \text{m/s})^2}{50 \, \text{N/m}}} \]
\[ x = \sqrt{\frac{0.5 \times 2.25}{50}} = \sqrt{\frac{1.125}{50}} = \sqrt{0.0225} \]

### Step 8: Calculate the maximum compression
\[ x = 0.15 \, \text{m} \]

### Conclusion
The maximum compression of the spring is **0.15 m**.