Question

in one dimensional collision the relative velocity before collision is equal to​

Answer 1

The negative of the relative velocity after collision, but the collision should be elastic.

Consider two masses m_1 and m_2 moving in the same straight line with velocities u_1 and u_2 respectively before collision where u_1 > u_2. After collision let them gain velocities v_1 and v_2 respectively.

[Refer https://brainly.in/question/12625162 for illustration]

According to law of conservation of linear momentum,

m_1 · u_1 + m_2 · u_2 = m_1 · v_1 + m_2 · v_2

m_1 · u_1 - m_1 · v_1 = m_2 · v_2 - m_2 · u_2

m_1 (u_1 - v_1) = m_2 (v_2 - u_2) → (1)

Since the collision is assumed to be elastic, we have total kinetic energy constant before and after collision. So,

[(m_1 (u_1)²) / 2] + [(m_2 (u_2)²)/2] = [(m_1 (v_1)²)/2] + [(m_2 (v_2)²/2]

In this equation, both sides are multiplied by 2.

(m_1 (u_1)²) + (m_2 (u_2)²) = (m_1 (v_1)²) + (m_2 (v_2)²

(m_1 (u_1)²) - (m_1 (v_1)²) = (m_2 (v_2)²) + (m_2 (u_2)²

m_1 [(u_1)² - (v_1)²] = m_2 [(v_1)² - (u_1)²]

m_1 (u_1 + v_1)(u_1 - v_1) = m_2 (v_2 + u_2)(v_2 - u_2) → (2)

Now, dividing (2) by (1), we get,

u_1 + v_1 = v_2 + u_2

u_1 - u_2 = v_2 - v_1

u_1 - u_2 = - (v_1 - v_2)

This equation states the relative velocity before collision is the negative of the relative velocity after collision.