A `16kg` block moving on a frictionless horizontal surface with a velocity of `4m//s` compresses an ideal spring and comes to rest. If the force constant of the spring be `100N//m`, then how much is the spring commpressed ?

Options

A. 1.7 m
B. 1.6 m
C. 1.4 m
D. 1.8 m

Answer

To solve the problem, we will use the principle of conservation of energy. The kinetic energy of the block will be converted into the potential energy stored in the spring when the block comes to rest.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Mass of the block, \( m = 16 \, \text{kg} \)
- Initial velocity of the block, \( v = 4 \, \text{m/s} \)
- Spring constant, \( k = 100 \, \text{N/m} \)

2. **Calculate the Initial Kinetic Energy (KE):**
The kinetic energy of the block when it is moving can be calculated using the formula:
\[ KE = \frac{1}{2} m v^2 \]
Substituting the values:
\[ KE = \frac{1}{2} \times 16 \, \text{kg} \times (4 \, \text{m/s})^2 = \frac{1}{2} \times 16 \times 16 = 128 \, \text{J} \]

3. **Set Up the Equation for Potential Energy (PE) of the Spring:**
When the block compresses the spring and comes to rest, all the kinetic energy is converted into potential energy stored in the spring:
\[ PE = \frac{1}{2} k x^2 \]
Where \( x \) is the compression of the spring.

4. **Equate the Kinetic Energy to the Potential Energy:**
Since the block comes to rest, we can set the kinetic energy equal to the potential energy:
\[ \frac{1}{2} k x^2 = KE \]
Substituting the values:
\[ \frac{1}{2} \times 100 \, \text{N/m} \times x^2 = 128 \, \text{J} \]

5. **Solve for \( x^2 \):**
\[ 50 x^2 = 128 \]
\[ x^2 = \frac{128}{50} = 2.56 \]

6. **Calculate \( x \):**
Taking the square root of both sides:
\[ x = \sqrt{2.56} = 1.6 \, \text{m} \]

### Final Answer:
The spring is compressed by \( 1.6 \, \text{m} \).

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