Question

scenario 4: 100% elastic collision between two particles of different mass and different initial velocity.
before collision
after collision

1. what is the relationship between the initial and final total momentums in scenario 4?
2. describe the motion of the particles before and after the collation in scenario 4.

Explanation:

Step1: Calculate initial momentum of particle 1

The formula for momentum is \(p = mv\). For particle 1, \(m_1=1.0\space kg\) and \(v_1 = 1.00\space m/s\). So \(p_1=m_1v_1=1.0\times1.00 = 1.0\space kg\cdot m/s\)

Step2: Calculate initial momentum of particle 2

For particle 2, \(m_2 = 0.5\space kg\) and \(v_2=- 0.50\space m/s\). Using \(p = mv\), \(p_2=m_2v_2=0.5\times(-0.50)=- 0.25\space kg\cdot m/s\)

Step3: Calculate total initial momentum

Total initial momentum \(P_{initial}=p_1 + p_2\). Substitute values: \(P_{initial}=1.0+( - 0.25)=0.75\space kg\cdot m/s\)

Step4: Use conservation of momentum (for elastic collision, \(P_{initial}=P_{final}\))

Since it is a \(100\%\) elastic collision, momentum is conserved. So total final momentum \(P_{final}=P_{initial}=0.75\space kg\cdot m/s\)

Step5: Use elastic - collision velocity formulas

The formulas for velocities after elastic collision are:
\(v_1'=\frac{(m_1 - m_2)v_1+2m_2v_2}{m_1 + m_2}\) and \(v_2'=\frac{(m_2 - m_1)v_2+2m_1v_1}{m_1 + m_2}\)
Substitute \(m_1 = 1.0\space kg\), \(m_2=0.5\space kg\), \(v_1 = 1.00\space m/s\), \(v_2=-0.50\space m/s\)
\(v_1'=\frac{(1.0 - 0.5)\times1.0+2\times0.5\times(-0.50)}{1.0 + 0.5}=\frac{0.5-0.5}{1.5}=0\space m/s\)
\(v_2'=\frac{(0.5 - 1.0)\times(-0.50)+2\times1.0\times1.00}{1.0 + 0.5}=\frac{0.25 + 2.0}{1.5}=\frac{2.25}{1.5}=1.5\space m/s\)

Step6: Calculate final momentum of particle 1

\(p_1'=m_1v_1'=1.0\times0 = 0\space kg\cdot m/s\)

Step7: Calculate final momentum of particle 2

\(p_2'=m_2v_2'=0.5\times1.5 = 0.75\space kg\cdot m/s\)

Answer:

Particle	Mass, \(m\) (\(kg\))	Initial Velocity, \(v\) (\(m/s\))	Initial Momentum, \(p\) (\(kg\cdot m/s\))	Final Velocity, \(v'\) (\(m/s\))	Final Momentum, \(p'\) (\(kg\cdot m/s\))
2	0.5	- 0.50	- 0.25	1.5	0.75
Total Initial Momentum (\(kg\cdot m/s\))	-	-	0.75	-	-
Total Final Momentum (\(kg\cdot m/s\))	-	-	-	-	0.75

The relationship between initial and final total momentum: The total initial momentum is equal to the total final momentum (\(P_{initial}=P_{final} = 0.75\space kg\cdot m/s\)) due to the law of conservation of momentum for elastic collisions.

After collision: Particle 1 stops (\(v_1'=0\space m/s\)), and particle 2 moves with a velocity of \(1.5\space m/s\) in the positive direction (opposite to its initial direction). Before collision: Particle 1 moves in the positive direction (\(v_1 = 1.00\space m/s\)), and particle 2 moves in the negative direction (\(v_2=-0.50\space m/s\))