Proving the conservation of momentum when two masses are not equal

[Stan2309]

Homework Statement

In this exercise we used steel and glass balls to calculate and prove the conservation of momentum in elastic collisions. I had to prove how momentum is conserved through displacements vectors and right triangles. All that stuff is easy and I got it. The last question is where I am stuck. The question states: When the mass of the 2marbles (glass and steel) are not the same, and they collide (the glass marble was initially at rest), why do you need to use the formula Pglass = (mglass/msteel) x Vglass (in the brackets the mass of the glass marble is divided by the mass of the steel marble, but we are not given any variables and are just required to prove or derive this formula)?

Homework Equations

I attempted to derive this formula through P total initial= P total final and the conservation of kinetic energy. 1/2m1v1i^2(squared) + 1/2m2v2i^2 = 1/2m1v1f^2 + 1/2m2v2f^2

The Attempt at a Solution

when i expand the formulas and substitute them into each other, I just end up with the equations for 1D elastic collisions v1f=(m1-m2/m1+m2)v1i and respectively v2f=(2m1/m1+m2)v1i
I spent the whole night going through all 3 of my physics textbooks and many websites online but couldn't find the way to prove or derive that formula. My teacher said it was a hard question but I never thought it would be this difficult. Oh by the way this is Gr.12 Academic physics.

 

[PeterO]

Homework Statement

In this exercise we used steel and glass balls to calculate and prove the conservation of momentum in elastic collisions. I had to prove how momentum is conserved through displacements vectors and right triangles. All that stuff is easy and I got it. The last question is where I am stuck. The question states: When the mass of the 2marbles (glass and steel) are not the same, and they collide (the glass marble was initially at rest), why do you need to use the formula Pglass = (mglass/msteel) x Vglass (in the brackets the mass of the glass marble is divided by the mass of the steel marble, but we are not given any variables and are just required to prove or derive this formula)?

Homework Equations

I attempted to derive this formula through P total initial= P total final and the conservation of kinetic energy. 1/2m1v1i^2(squared) + 1/2m2v2i^2 = 1/2m1v1f^2 + 1/2m2v2f^2

The Attempt at a Solution

when i expand the formulas and substitute them into each other, I just end up with the equations for 1D elastic collisions v1f=(m1-m2/m1+m2)v1i and respectively v2f=(2m1/m1+m2)v1i
I spent the whole night going through all 3 of my physics textbooks and many websites online but couldn't find the way to prove or derive that formula. My teacher said it was a hard question but I never thought it would be this difficult. Oh by the way this is Gr.12 Academic physics.

In this experiment you are measuring velocities [by referring to displacement in a fixed time]

You are then trying to show conservation of momentum.

The only measure you have is velocity, but momentum is the product of mass and velocity.

When you do the experiment with two equal steel balls, the mass of a steel ball is effectively just a scaling factor for the velocity - so you can do a diagram of velocity vectors - in arbitrary units - to show momentum is conserved.

When you use balls of different masses [steel and glass], to merely use the velocity of each ball is not suitable.
Since velocity is measured in arbitrary units - displacement / some time (perhaps the time taken to fall a set distance) - and mass is effectively a "scaling factor", we have to match those scaling factors. By using M_glass/ M_steel that is achieved.
or example; if the glass ball is 1/3rd the mass of the steel ball, it will be traveling 3-times a s fast for an equivalent momentum value.

Hope that all makes sense.

Peter

 

[Stan2309]

yeah i get what we're doing and all, but my teacher want me to derive this formula. I'm suspecting it has something to do with the glass/steel ratio and the displacement vectors. Need mathematical proof D:

 

[PeterO]

yeah i get what we're doing and all, but my teacher want me to derive this formula. I'm suspecting it has something to do with the glass/steel ratio and the displacement vectors. Need mathematical proof D:

You will never be able to prove it!

The formula you wrote was:

Pglass = (mglass/msteel) x Vglass

LHS has units of momentum; because it is momentum.

RHS has units of velocity; because the mass units cancel out.

 

[asteriatic]

Firstly, I want to commend you for your efforts in trying to derive and understand this formula. It shows your dedication and determination as a scientist.

Now, to answer your question, the formula Pglass = (mglass/msteel) x Vglass is used in cases where the two masses are not equal because it takes into account the difference in masses between the two objects. In an elastic collision, the total momentum before and after the collision must be equal. However, in the case where the masses are not equal, the velocities of the two objects after the collision will also be different. This is where the formula comes in.

By using the formula Pglass = (mglass/msteel) x Vglass, we are able to calculate the velocity of the glass marble after the collision, taking into account the difference in masses between the glass and steel marbles. This is necessary in order to satisfy the conservation of momentum principle.

In order to derive this formula, you can use the momentum conservation equation, as you have already attempted. However, when substituting the velocities, you should use the formula for conservation of kinetic energy for elastic collisions, which is (1/2)m1v1i^2 + (1/2)m2v2i^2 = (1/2)m1v1f^2 + (1/2)m2v2f^2. From here, you can solve for v1f and substitute it into the momentum conservation equation. After some algebraic manipulation, you should arrive at the formula Pglass = (mglass/msteel) x Vglass.

I hope this explanation helps you understand the reasoning behind the formula and how to derive it. Keep up the good work and never stop questioning and seeking knowledge as a scientist.