Question

Two blocks of masses 20 kg and 0.50 kg are attached at the two ends of a
compressed spring. The elastic potential energy stored in the spring is 10 J. Find the
velocities of the blocks if the spring delivers its energy to the blocks when released. ​

Answer 1

According to the center-of-mass theorem (https://link.springer.com/chapter/10.1007%2F978-3-642-57234-0_9), the spring delivers the equal amount of momentum (and energy) to the both blocks. Thus, each one will obtain $10J/2 = 5J$.

All this energy will be spend on the increasing the kinetic energy of the blocks. By definition, the kinetic energy is:

$$K = \dfrac{mv^2}{2}$$

where $m$ is the mass of the block and $v$ is it's velocity. Expressing the velocity, obtain:

$$v = \sqrt{\dfrac{2K}{m}}$$

$v_1 =$ Thus, let's substitute numerical values.

For the first block:

$$v_1 = \sqrt{\dfrac{2K}{m_1}} = \sqrt{\dfrac{2\cdot 5J}{20kg}} \approx 0.707m/s$$

For the second block:

$$v_2 = \sqrt{\dfrac{2K}{m_2}} = \sqrt{\dfrac{2\cdot 5J}{0.5kg}} \approx 4.47m/s$$

Answer. v1 = 0.707 m/s, v2 = 4.47 m/s.