- #1
Nirmal Padwal
- 41
- 2
- Homework Statement
- Use a three-particle interaction to show that mass is an additive quantity.
- Relevant Equations
- ## m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2##.
I already have referred to the solution to this problem. But the way I originally solved the problem is completely different from how the available solution proceeds. I wish to know if my solution is right or wrong.
My Solution:
Consider three particles undergoing one-dimensional motion all moving in the same direction. Let the leftmost particle be ##A## whose mass and initial velocity are ##m_1 ## and ## u_1## respectively. Let the particle in between be ##B## whose mass and initial velocity are ##m_2 ## and ## u_2## respectively. Let the rightmost particle be ##C## whose mass and initial velocity are ##m_3 ## and ## u_3## respectively. Consider that ## u_1 > u_2 > u_3## and the distances between ##A##, ##B## and ##C## are such that ##A## and ##B## collide first and the combined mass then collides with ##C##. We assume inelastic collision.
By conservation of linear momentum (not even mentioned in the actual solution), momentum before and after collision between ##A## and ##B## are related by $$ m_1 u_1 + m_2 u_2 = m_1 u + m_2 u = (m_1 +m_2) u = m_c u $$ where u is the velocity of combined mass
Similarly, by conservation of linear momentum, momentum before and after collision between combined mass ##m_c## and ##C## are related by $$ m_c u + m_3 u_3 = m_c u' + m_3 u' = (m_c +m_3) u' = (m_1 + m_2+m_3) u' = m u' $$ where u' is the velocity of combined mass of all three particles.
Thus we have ## m u' = (m_1 + m_2+m_3) u'## or ## m = m_1 +m_2 +m_3##. Hence mass is an additive quantity
Is this solution correct?
My Solution:
Consider three particles undergoing one-dimensional motion all moving in the same direction. Let the leftmost particle be ##A## whose mass and initial velocity are ##m_1 ## and ## u_1## respectively. Let the particle in between be ##B## whose mass and initial velocity are ##m_2 ## and ## u_2## respectively. Let the rightmost particle be ##C## whose mass and initial velocity are ##m_3 ## and ## u_3## respectively. Consider that ## u_1 > u_2 > u_3## and the distances between ##A##, ##B## and ##C## are such that ##A## and ##B## collide first and the combined mass then collides with ##C##. We assume inelastic collision.
By conservation of linear momentum (not even mentioned in the actual solution), momentum before and after collision between ##A## and ##B## are related by $$ m_1 u_1 + m_2 u_2 = m_1 u + m_2 u = (m_1 +m_2) u = m_c u $$ where u is the velocity of combined mass
Similarly, by conservation of linear momentum, momentum before and after collision between combined mass ##m_c## and ##C## are related by $$ m_c u + m_3 u_3 = m_c u' + m_3 u' = (m_c +m_3) u' = (m_1 + m_2+m_3) u' = m u' $$ where u' is the velocity of combined mass of all three particles.
Thus we have ## m u' = (m_1 + m_2+m_3) u'## or ## m = m_1 +m_2 +m_3##. Hence mass is an additive quantity
Is this solution correct?