Problem 8-34 Elastic collision - Part 5 - A
A superball of mass \(22\; g\) rolls with a speed of \(3.5\; m/s\) toward another (stationary) superball of mass \(27\; g\). If the balls have a head-on elastic collision, what are the velocities magnitudes and directions) of the balls after the collision?
[Ans. \(3.1\; m/s\) forward and \(0.4\; m/s\) backward]
Accumulated Solution
\(m_1\; v_1 = m_1\; v{_1}'+ m_2\; v{_2}' \\ 3.5 = v{_1}' + 1.227\; v{_2}' \; \text{Eqn. #1}\\ m_1\; v{_1}{^2} = m_1\; v{_1}'{^2} + m_2\; v{_2}'{^2} \\ 3.5^2 = v{_1}'{^2} + 1.227\;v{_2}'{^2}\; \text{Eqn. #2}\)
\(\text{Eqn.#1} \; v{_1}' = 3.5 - 1.227\; v{_2}' \\ v{_1}'{^2} =3.52- 8.589\; v{_2}'+ 1.506\; v{_2}'{^2}\)
Substitute in \(\text{Eqn.#2}\):
\(3.52 = 3.52- 8.589\; v{_2}'+ 1.506\; v{_2}'{^2} + 1.227\; v{_2}'{^2}\)
Notice that the \(3.5^2\) subtracts away and then one power of \(v{_2}'\) can be cancelled from the remaining terms leaving:
\(0 = 1.506\; v{_2}' - 8.589 + 1.227\; v{_2}'\)
which can be solved for \(v{_2}'\)
Do this before continuing.