E1.5 Elastic Collision in 1 - Dimensional:

If the colliding bodies move along the same straight path before and after the collision, it is said to be one - dimensional collision (Principal of PHYSICS).

Consider two bodies of masses $m_1$ and $m_2$ moving with initial velocities $u_1$ and $u_2$ (such that $u_1 > u_2$) in a same direction. Let after the collision velocity of the bodies change into $v_1$ and $v_2$ in a same direction.

According to principle of Conservation of Linear Momentum,

$m_1\,u_1 + m_2\,u_2 = m_1\,v_1 + m_2\,v_2$ .......... (i)

$m_1\,(u_1 - v_1) = m_2\,(v_2 - u_2)$ .......... (ii)

According to the principle of Conservation of Kinetic Energy,

$\frac{1}{2}\,m_1\,u{_1}^2 + \frac{1}{2}\,m_2\,u{_2}^2 = \frac{1}{2}\,m_1\,v{_1}^2 + \frac{1}{2}\,m_2\,v{_2}^2$

or,  $m_1\,(u{_1}^2 - v{_1}^2) = m_2\,(v{_2}^2 - u{_2}^2)$

or,       $m_1\,(u{_1} + v{_1})(u{_1} - v{_1}) = m_2\,(v{_2} + u{_2})(v{_2} - u{_2})$ .......... (iii)

Dividing equation (iii) by equation (ii), then we get

$u_1 + v_1 = u_2 + v_2$

or, $u_1 - u_2 = v_2 - v_1$ .......... (iv)

This equation (iv) shows that: "In Perfectly elastic collision the relative speed of approach ($u_1 - u_2$) is equal to the relative speed of separation ($v_2 - v_1$)".

» Now, we have to calculate the final velocity of bodies $v_1$ and $v_2$:

From equation (iv),        $v_2 = u_1 - u_2 + v_1$ .......... (v)

Substituting the value of $v_2$ in equation (i),

$m_1\,u_1 + m_2\,u_2 = m_1\,v_1 + m_2\,(u_1 - u_2 + v_1)$

⇒ $v_1 = \frac{(m_1 - m_2)\,u_1 + 2\,m_2\,u_2}{m_1 + m_2}$ .......... (vi)

Similarly,

⇒ $v_2 = \frac{(m_2 - m_1)\,u_2 + 2\,m_1\,u_1}{m_1 + m_2}$ .......... (vii)

Play with this: Elastic Collision (1 - D Calculator).

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