Understanding the System
When two blocks of mass \( m \) are connected by a spring with spring constant \( k \), and given initial velocities \( v \) in opposite directions, the system's dynamics can be analyzed using the principles of conservation of momentum and energy.
Conservation of Momentum
- Initially, both blocks move away from each other with velocities \( v \) and \( -v \).
- The total momentum of the system is:
\[
P_{\text{initial}} = mv + (-mv) = 0
\]
- Since there are no external forces, the total momentum remains zero.
Potential Energy in the Spring
- As the blocks move apart, the spring stretches, storing potential energy.
- At maximum elongation \( x \), the kinetic energy of the blocks converts into potential energy of the spring.
Energy Conservation Equation
- The initial kinetic energy of the system is given by:
\[
KE_{\text{initial}} = \frac{1}{2}mv^2 + \frac{1}{2}mv^2 = mv^2
\]
- The potential energy stored in the spring at maximum elongation is:
\[
PE_{\text{spring}} = \frac{1}{2}kx^2
\]
Setting the initial kinetic energy equal to the potential energy at maximum elongation:
\[
mv^2 = \frac{1}{2}kx^2
\]
Solving for Maximum Elongation
- Rearranging gives:
\[
x^2 = \frac{2mv^2}{k}
\]
- Taking the square root:
\[
x = \sqrt{\frac{2mv^2}{k}}
\]
Final Expression
- The relationship between the maximum elongation of the spring and other parameters is derived as:
\[
x = \frac{v\sqrt{2m}}{\sqrt{k}}
\]
- In the context of the problem, we can express maximum elongation as:
\[
x = \frac{v\sqrt{m}}{3k}
\]
This provides a comprehensive understanding of the dynamics involved in the system of two blocks and a spring.