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 can follow these steps:

### Step 1: Convert the speed from km/h to m/s
The speed of the car is given as 18 km/h. We need to convert this to meters per second (m/s).

\[ \text{Speed in m/s} = \text{Speed in km/h} \times \frac{5}{18} \]

Substituting the value:

\[ \text{Speed in m/s} = 18 \times \frac{5}{18} = 5 \, \text{m/s} \]

### Step 2: Write down the equations for kinetic energy and potential energy
At maximum compression of the spring, the kinetic energy of the car will be equal to the potential energy stored in the spring.

- Kinetic Energy (KE) of the car:
\[ KE = \frac{1}{2} m v^2 \]

- Potential Energy (PE) of the spring:
\[ PE = \frac{1}{2} k x_m^2 \]

Where:
- \( m = 1000 \, \text{kg} \) (mass of the car)
- \( v = 5 \, \text{m/s} \) (speed of the car)
- \( k = 6.25 \times 10^3 \, \text{N/m} \) (spring constant)
- \( x_m \) = maximum compression of the spring

### Step 3: Set the kinetic energy equal to the potential energy
At maximum compression, we have:

\[ \frac{1}{2} m v^2 = \frac{1}{2} k x_m^2 \]

### Step 4: Simplify the equation
We can cancel \(\frac{1}{2}\) from both sides:

\[ m v^2 = k x_m^2 \]

### Step 5: Solve for maximum compression \( x_m \)
Rearranging the equation gives:

\[ x_m^2 = \frac{m v^2}{k} \]

Taking the square root:

\[ x_m = \sqrt{\frac{m v^2}{k}} \]

### Step 6: Substitute the known values
Now, substituting the values of \( m \), \( v \), and \( k \):

\[ x_m = \sqrt{\frac{1000 \times (5)^2}{6.25 \times 10^3}} \]

Calculating the values:

\[ x_m = \sqrt{\frac{1000 \times 25}{6250}} = \sqrt{\frac{25000}{6250}} = \sqrt{4} = 2 \, \text{m} \]

### Conclusion
The maximum compression of the spring is \( 2 \, \text{m} \).

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