Elastic Collision Formula: Solving for Final Velocity and Mass Ratios

AI Thread Summary
In a perfectly elastic collision, an object of mass m1 traveling with initial velocity v1i collides with a much larger mass m2 moving at velocity v2i. For m1 to stop completely after the collision, v1i must equal 2v2i. If v1i is greater than this value, m1 will rebound, while if it is smaller, m1 will come to a stop without rebounding. The discussion includes attempts to derive the final velocities using conservation of energy and momentum, but the user encounters mathematical errors. Clarification on the dimensional validity of the velocity equation is also sought.
Puddles
Messages
12
Reaction score
0
Homework Statement

An object of mass m1 traveling with velocity v1i has a perfectly elastic collision in which it rear ends and object of mass m2 (m2>>m1) traveling with velocity v2i. How must the velocity v1i relate to v2i if the mass m1 is to stop in its tracks (v1f=0)? What happens if velocity v1i is greater than this? If it is smaller?

Relevant equations
KE = .5mv^2
P = mv

The attempt at a solution
Cons Energy
.5m1v1i^2 + .5m2v2i^2 = .5m2v2f^2

V2f = sq.rt(( m1v1i^2 + m2v2i^2 )/(m2))

Cons Momentum
m1v1i + m2v2i = m2v2f

V2f = ( m1v1i + m2v2i )/(m2)

Set equal to each other, but my answer keeps getting more complex? It's a math error, but I'm not sure what it is…

I get to here:(m1^2v1i^2)+(2m1v1im2v2i)+(m2^2v2i^2) = (m1v1i^2)+(m2v2i^2)

Can anyone help me continue to work this out? I'm frustrated because this is a simple problem but I can't get it.
 
Physics news on Phys.org
Okay, I think I found my error, I've worked out that v1i = (m2(1 - 2v2i))/(m1), how can I plug this back into check it? I'm struggling to find a way to do so but I know there must be a way…
 
Puddles said:
Okay, I think I found my error, I've worked out that v1i = (m2(1 - 2v2i))/(m1), how can I plug this back into check it? I'm struggling to find a way to do so but I know there must be a way…
Is there a typo there? 1-velocity is dimensionally invalid.
 
Thread 'Variable mass system : water sprayed into a moving container'
Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...

Similar threads

Back
Top