Collision of Two Hard Spheres: Solving the Center of Mass Problem

[hkyriazi]

I'm having trouble figuring out what seems to be a simple problem involving a center of mass collision of two hard spheres. Sphere #1 has mass M, and Sphere #2 has a mass of M/2. Sphere 1 is moving at velocity V, while Sphere 2 is stationary. What are their speeds after the collision? My problem is, that to conserve both momentum and kinetic energy, it seems that Sphere 2 must have a resulting speed equal to 4V/3 (and Sphere 1 a speed of V/3), which seems impossible. How could the impact from an object moving at only speed V lead an object initially at rest to recoil at an even greater speed (1.333V)?

For Momentum, this yields:

MV = (M/2)(4V/3) + MV/3 = 4/6 MV + 1/3 MV = 2/3 MV+ 1/3 MV

For Kinetic Energy:

1/2 MV² = 1/2 (M/2) (4V/3)² + 1/2 M(V/3)²
=M/4 (16/9) V² + 1/18 MV² = 4/9 MV² + 1/18 MV² = 9/18 MV² = 1/2 MV²

 

[cesiumfrog]

Firstly, you only need to conserve momentum.

In a way, momentum conservation is "better" or "stronger" than energy conservation. Sure, the latter is true in principle, but in practice either something blows up to release extra energy, or some kinetic energy is lost to heat, noise, etc. It's more obvious when a system looses significant momentum (big bits fly off).

Secondly, if you don't believe big thing hitting small thing can make the small thing go really fast, balance a tennis ball on an indoor soccerball (or a basketball) and drop them on the ground.

 

[hkyriazi]

Thanks. I actually checked out the relationship between golf clubhead speed and ball speed yesterday, and did indeed determine that a more massive object can impact a stationary object and make it move faster than its own speed. (That webpage gave a typical ball speed as 140 mph, and clubhead speed as 95 mph.)

I think what was throwing off my intuition was that I must've been imagining aspects of an inelastic collision, where the objects would move at the same speed while they were connected, and then, since I was also imagining no compression, there wouldn't be any additional recoil momentum/speed added. In that case, the impacting object would just be "throwing" the impacted object, not bouncing off it. Instead, I should've been thinking of an instantaneous recoil.