What happens if you throw 2 balls of equal mass and equal velocity at each other and they combine into one object?

Submitted by minecraftmanyt t3_11vsibw in askscience

So for them to combine into one object they'll have to be perfectly inelastic, so that's the assumption.

Logically, they would hit eachother then perfectly stop, because no other outcome makes sense.

However, doing the math for sum of work on 2 balls with 1 kg and 1 m/s each:

W1=0.5*m_1*v_1^2=0.5*1*1^2=0.5

W2=0.5*m_2*v_2^2=0.5*1*1^2=0.5

Wnet=0.5+0.5=1=1/2*m_net*v_net^2=1/2*2*v_net^2

v_net=0.707.

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How is this possible? How can they have a velocity? Which direction would it be going? Logically, they would stop in place.

Is it therefore the case that this equation does not work in this situation? Is it because this equation ignores loss to heat and sound etc. and normally that loss is somewhat negligible? I'm a little confused.

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Comments

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Weed_O_Whirler t1_jcysohq wrote

While it's true energy is always conserved, kinetic energy is only conserved in perfectly elastic collisions. So, you are on the correct path at the end- inelastic collisions kinetic energy is converted into other types of energy, like heat, sound, deforming the bodies, etc.

If kinetic energy was conserved in the collision, then you would have the situation where the two objects bounced off of each other, and went in the opposite direction, at the same speed they came in.

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Gedankensortieren t1_jcyuy4a wrote

You should use conservation of momentum. This is always conserved.

Regarding kinetic energy: It is conservation of energy, not conservation of kinetic energy. In an inelastic collision the kinetic energy is transformed into a different kind of energy

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mutandis57 t1_jd018fx wrote

> ... > > 1=1/2*2*v_net^2 > > v_net=0.707

You missed the extra factor of "*2" at the end and took the square root of 1/2 by mistake. Had you taken the square root of "1/2*2" you would have gotten v_net=1, i.e. the speed of the two balls together is the same as each ball separately. But that's because, as other commenters pointed out, you used conservation of energy for a perfectly-elastic collision. Using conservation of momentum in a perfectly-inelastic collision v_net will be zero and all the kinetic energy will be destroyed (converted into thermal energy of the atoms inside the ball bouncing around randomly).

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minecraftmanyt OP t1_jdoqfxc wrote

Oh yeah I didn't even notice that, yeah it should have been 1 thanks!

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bad_bird_karamaru t1_jczvl4k wrote

As the others have mentioned, momentum is conserved in a perfectly inelastic collision, but kinetic energy is not. Assuming that not all of the missing energy is lost to heat, sound or the deformation of the balls, a force will be present to hold the two balls together, so that they stick. The missing kinetic energy is converted to the potential energy associated with the force, which you might call the "binding energy" for the system.

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