Conservation of linear momentum and kinetic energy

[knowNothing23]

A particle of mass m1 traveling with a speed v makes a head-on elastic
collision with a stationary particle of mass m2. In which scenario will the largest
amount of energy be imparted to the particle of mass m2? (a) m2 < m1,
(b) m2 = m1, (c) m2 > m1, (d) None of the above.

Why do I need to differentiate the ratio of the kinetic energy and equate it to 0 to find the answer?

 

[PeterO]

A particle of mass m1 traveling with a speed v makes a head-on elastic
collision with a stationary particle of mass m2. In which scenario will the largest
amount of energy be imparted to the particle of mass m2? (a) m2 < m1,
(b) m2 = m1, (c) m2 > m1, (d) None of the above.

Why do I need to differentiate the ratio of the kinetic energy and equate it to 0 to find the answer?

In an elastic collision, kinetic energy is conserves - and since KE is NOT a vector, we can't have a positive amount with one body and a negative amount with the other.

For the second body to have maximum KE, we want the first body to have minimum KE - and you can't get less than zero.

Which of the mass ratios will have m1 stopping?

EDIT: we are looking for maximum KE, not maximum velocity for m2.

 

[knowNothing23]

I have to use the first and only initial kinetic energy of the moving mass1 and compare it to the final kinetic energy of mass2 or mass that doesn't move to find the conditions, where mass1 imparts the largest kinetic energy to mass2.

I've seen the solution, but can't figure it out, why is the ratio of: KfinalOfMass2/KinitialOfMass1 is equated to O and differentiated.

 

[PeterO]

I have to use the first and only initial kinetic energy of the moving mass1 and compare it to the final kinetic energy of mass2 or mass that doesn't move to find the conditions, where mass1 imparts the largest kinetic energy to mass2.

I've seen the solution, but can't figure it out, why is the ratio of: KfinalOfMass2/KinitialOfMass1 is equated to O and differentiated.

If that ratio had been first differentiated and then equated to zero, it would be part of a standard maximum/minimum problem.

 

[knowNothing23]

Could you explain that rule of maximum and minimum problem? I'm familiar with it's use to find stable and unstable equilibrium.

 

[PeterO]

Could you explain that rule of maximum and minimum problem? I'm familiar with it's use to find stable and unstable equilibrium.

At a local maximum or minimum, the derivative is equal to zero. Once you have identified a point you need to verify it is the max or min you were seeking.

 

[knowNothing23]

How do you verify that? And, why is it that one can find that with the derivative?

 

[bumpkin]

How do you verify that? And, why is it that one can find that with the derivative?

derivative can also be described as rate of change. Rate of change is zero at any point in the original function where there is a minimum or a maximum. In this case, you are trying to maximise the energy transfer, or the point where the ratio of the kinetic energy of the two particles is a maximum. You could plot the ratios on a graph and look for the maximum that way' but it is easier to differentiate, set to zero, solve for mass ratio.