A car of mass 1000 kg moving with a speed `18 km h^(-1)` on a smooth road and colliding with a horizontally mounted spring of spring constant `6.25 xx10^3 N m^(-1)`. The maximum compression of the spring is

Options

A. 1 m
B. 2 m
C. 3 m
D. 4 m

Explanation

To solve the problem of finding the maximum compression of the spring when a car collides with it, we will use the principle of conservation of energy. Here’s a step-by-step solution:

### Step 1: Convert the speed from km/h to m/s
The speed of the car is given as 18 km/h. To convert this to meters per second (m/s), we use the conversion factor:
\[ 1 \text{ km/h} = \frac{1}{3.6} \text{ m/s} \]
Thus,
\[ v = 18 \text{ km/h} \times \frac{1 \text{ m/s}}{3.6 \text{ km/h}} = 5 \text{ m/s} \]

### Step 2: Write down the kinetic energy of the car
The kinetic energy (KE) of the car can be calculated using the formula:
\[ KE = \frac{1}{2} mv^2 \]
Where:
- \( m = 1000 \text{ kg} \) (mass of the car)
- \( v = 5 \text{ m/s} \) (speed of the car)

Substituting the values:
\[ KE = \frac{1}{2} \times 1000 \text{ kg} \times (5 \text{ m/s})^2 = \frac{1}{2} \times 1000 \times 25 = 12500 \text{ J} \]

### Step 3: Write down the potential energy stored in the spring
The potential energy (PE) stored in the spring at maximum compression can be expressed as:
\[ PE = \frac{1}{2} K X^2 \]
Where:
- \( K = 6.25 \times 10^3 \text{ N/m} \) (spring constant)
- \( X \) is the maximum compression of the spring.

### Step 4: Apply the conservation of energy principle
According to the conservation of energy, the kinetic energy of the car will be equal to the potential energy stored in the spring at maximum compression:
\[ KE = PE \]
Thus,
\[ \frac{1}{2} mv^2 = \frac{1}{2} K X^2 \]

### Step 5: Cancel the common factors and solve for \( X \)
Cancelling \( \frac{1}{2} \) from both sides, we have:
\[ mv^2 = K X^2 \]
Rearranging for \( X \):
\[ X^2 = \frac{mv^2}{K} \]
Taking the square root:
\[ X = \sqrt{\frac{mv^2}{K}} \]

### Step 6: Substitute the known values
Substituting \( m = 1000 \text{ kg} \), \( v = 5 \text{ m/s} \), and \( K = 6.25 \times 10^3 \text{ N/m} \):
\[ X = \sqrt{\frac{1000 \times (5)^2}{6.25 \times 10^3}} = \sqrt{\frac{1000 \times 25}{6250}} = \sqrt{4} = 2 \text{ m} \]

### Final Answer
The maximum compression of the spring is:
\[ \boxed{2 \text{ m}} \]