Proof on two dimensional elastic collision.

In summary, when a ball with mass m collide with another ball with equal mass as m1, the angle between the masses would be 90' after the collision. However, there is no mathematical proof to show this result.
  • #1
mysqlpress
83
1
When a ball with mass m collide with another ball with equal mass as m=> m1 at rest, the mathematical proof shows that after the collision, the angle between two masses would be 90'

I know that when m> m1 , angle <90'
when m<m1 , angle >90'

but there is no mathematical proof to show this result...
...
does anyone help me solve this problem?
 
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  • #2
Can you prove the first result? (For equal masses.)
 
  • #3
making use of KE conserved and momentum conserved...

mv=m1v1cosx+mv2cosy- (1)
v=v1cosx+v2cosy
0=m1v1sinx-mv2siny - (2)
0=v1sinx-v2siny
(1)^2+(2)^2

v^2 = (v1)^2+(v2)^2 +2v1v(cosxcosy-sinysinx)

KE conserved
mv^2 =m1v1^2+mv2^2
v^2=v1^2+v2^2
cos(x+y) = 90'
 
  • #4
Good. Now redo it without canceling the masses. You'll get an expression for cos(x+y) that will depend on m - m1.
 
  • #5
Doc Al said:
Good. Now redo it without canceling the masses. You'll get an expression for cos(x+y) that will depend on m - m1.
Thanks ! I've proven that. :)
 
  • #6
Could you please describe the work in more detail? I understand the entire concept entirely, and even matched up with a lot of the math you did, but I didn't understand the combining you did in the middle and how the angle 90 just appeared out of nowhere.
 
  • #7
mysqlpress said:
making use of KE conserved and momentum conserved...

mv=m1v1cosx+mv2cosy- (1)
v=v1cosx+v2cosy
0=m1v1sinx-mv2siny - (2)
0=v1sinx-v2siny
(1)^2+(2)^2

v^2 = (v1)^2+(v2)^2 +2v1v(cosxcosy-sinysinx)

KE conserved
mv^2 =m1v1^2+mv2^2
v^2=v1^2+v2^2
cos(x+y) = 90'

I understand everything except what is in bold. First, you are finding the momentum along the X. Then finding the momentum along the Y. Then somehow combine them in the equation for the conservation of energy. I have all of that already, but I don't understand the combining itself and don't understand the bolded areas
 
  • #8
NvM. I got it. Law of cosines and sum difference formulas.

Ø1 + ø2 = 90,

so it is safe to assume that :

cos(Ø1 + ø2) = 0

since cos(90) = 0. Now, for the actual proof, start by using the formula for conservation of kinetic energy:

K1i + K2i = K1f + K2f.

After simplifying, you will get:

V1i^2 = V1f^2 + V2f^2.

According to law of cosines,

c^2 = a^2 + b^2 -2(ab)cosC.

CosC is The same value as cos(ø1 + ø2), so

c^2 = a^2 + b^2 -2(ab)cos
 
  • #9
44r0n0wnz said:
NvM. I got it. Law of cosines and sum difference formulas.
Good! (Sorry I didn't have time to respond earlier.)
 

Related to Proof on two dimensional elastic collision.

What is a two dimensional elastic collision?

A two dimensional elastic collision is a type of collision between two objects where both the momentum and kinetic energy are conserved. This means that after the collision, the total momentum and total kinetic energy of the system remains the same.

What is the formula for calculating the velocities after a two-dimensional elastic collision?

The formula for calculating the velocities after a two-dimensional elastic collision is v1 = (m1-m2)v1i/(m1+m2) + (2m2v2i)/(m1+m2) and v2 = (2m1v1i)/(m1+m2) + (m2-m1)v2i/(m1+m2) where v1i and v2i are the initial velocities of the objects and m1 and m2 are the masses of the objects.

What is the difference between an elastic and inelastic collision?

In an elastic collision, both momentum and kinetic energy are conserved. However, in an inelastic collision, only momentum is conserved while kinetic energy is lost. This means that after an inelastic collision, the objects involved will have a lower total kinetic energy compared to before the collision.

How do you determine if a collision is elastic or inelastic?

You can determine if a collision is elastic or inelastic by calculating the total kinetic energy of the system before and after the collision. If the total kinetic energy remains the same, then the collision is elastic. If the total kinetic energy decreases, then the collision is inelastic.

What real-life examples demonstrate two dimensional elastic collisions?

Some real-life examples of two dimensional elastic collisions include billiard balls colliding on a pool table, two hockey players colliding on the ice, and two cars colliding on a flat surface with no external forces acting on them.

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