2 Cars Colliding with same mass

[YinHoNg]

Hiya. The problem I'm trying to solve is this:

• I have two cars, each with the same mass.
• Speed before Collision of Car1 = Xm/s
• Speed before Collision of Car2 = Ym/s

To solve the problem of what speeds the two cars have after an ELASTIC collision, i used simultaneous equations.
My process was:

1. make a K.E equation
2. Make a momentum equation
3. Solve for V1, and then thus V2

From my workings, i seem to have found that if the masses of the two vehicles are equal, then in an elastic collision; the speeds afterwards of the two vehicles will just swap.

e.g.

1. Both cars' mass = 2Kg
2. Speed of Car1 before collision = +4m/s
3. Speed of Car2 before collision = -2m/s
4. Speed of Car1 after collision = -2m/s = Speed of Car2 before collision
5. Speed of Car2 after collision = +4m/s = Speed of Car1 before collision

Will that be the same for all collisions where the mass is the same and the collision is elastic?

 

[Doc Al]

Will that be the same for all collisions where the mass is the same and the collision is elastic?

Indeed it will. (Good thinking!) Why not try to prove this for yourself in general using the same method that you used above, but using symbols, not plugging in specific numbers. You'll need to solve these two equations simultaneously:
$$mv_1 + mv_2 = mv'_1 + mv'_2$$
$$mv_1^2 + mv_2^2 = mv'_1^2 + mv'_2^2$$

 

[quicknote]

I can confirm that your findings are correct. In an elastic collision between two objects with the same mass, the velocities will swap after the collision. This is due to the conservation of momentum and energy. In an elastic collision, the total kinetic energy of the system is conserved, meaning that the total energy before the collision is equal to the total energy after the collision. This also applies to the momentum of the system, where the total momentum before the collision is equal to the total momentum after the collision. Therefore, in this scenario, the velocities of the two cars will swap after the collision. This will hold true for all collisions where the mass is the same and the collision is elastic. Great job on using simultaneous equations to solve this problem!