How Do Elastic Collisions Affect Velocities and Center of Mass Movement?

[DatAshe]

Homework Statement

A ball of mass 0.206 kg with a velocity of 1.54 m/s meets a ball of mass 0.291 kg with a velocity of -0.396 m/s in a head-on, elastic collision.
(a) Find their velocities after the collision.
1f = m/s
2f = m/s
(b) Find the velocity of their center of mass before and after the collision.
cm, before = m/s
cm, after = m/s

Homework Equations

conservation of momentum? p=mv, pf=pi+I, KEf=KEi (because its elastic), Vcm=Ʃmivi/Ʃmi

The Attempt at a Solution

I have spent hours fiddling around with equations and I can't get anything to result in a correct answer. I am stumped and frustrated. This is past homework and I am just looking for an explanation on how I get from this information to the final answers. Step by step would be much appreciated, especially if you use calculus because I am lost and I am only to derivatives in calc I so far. Thank you for any and all help!

 

[haruspex]

Yes, conservation of energy and of momentum.
What is the total momentum before? Total energy before?
In terms of the unknown final velocities, what are the energy and momentum after collision?
What equations do you get?

 

[DatAshe]

wouldn't the before and after be the same do to conservation of momentum and energy?
so mivi^2=mfvf^2 and mivi=mfvm?

 

[AJ Bentley]

You have to calculate the total momentum and energy.

That is the momentum of both balls added together. The sum total is the same before and after the collision. m1*v1i + m2*v2i = m1*v1f + m2*v2f

Same with energy.

That gives you a couple of equations with two unknowns (v1f and v2f) to solve.

 

[Nickie]

I would approach this problem by first defining the scenario and the variables involved. In this case, we have two balls of different masses colliding in a head-on, elastic collision. The masses of the balls are 0.206 kg and 0.291 kg, and their initial velocities are 1.54 m/s and -0.396 m/s respectively. We are asked to find the final velocities of the balls and the velocity of their center of mass before and after the collision.

Next, I would use the principles of conservation of momentum and conservation of kinetic energy to solve this problem. These principles state that in an isolated system (where there are no external forces acting), the total momentum and total kinetic energy remain constant before and after a collision.

First, we can calculate the total momentum before the collision. We can do this by multiplying the mass of each ball by its initial velocity and adding them together. In this case, we have:

initial momentum = (0.206 kg)(1.54 m/s) + (0.291 kg)(-0.396 m/s) = 0.318 kgm/s

Since this is an elastic collision, the total momentum after the collision will also be equal to 0.318 kgm/s. Now, we can use this fact to solve for the final velocities of the balls.

We can define the final velocities as v1f and v2f for the first and second ball respectively. Using the conservation of momentum principle, we can set up the following equation:

m1v1i + m2v2i = m1v1f + m2v2f

Substituting the values we know, we get:

(0.206 kg)(1.54 m/s) + (0.291 kg)(-0.396 m/s) = (0.206 kg)v1f + (0.291 kg)v2f

Solving for v1f and v2f, we get:

v1f = (0.206 kg)(1.54 m/s) + (0.291 kg)(-0.396 m/s) - (0.291 kg)(0.206 kg)(-0.396 m/s) = 0.872 m/s

v2f = (0.206 kg)(1.54 m/s) + (0.291 kg)(-0.396 m/s) - (0.206