[Julia]

Julia asked the Naked Scientists:
According to Newton\'s third law, \"to every action there is an equal and opposite reaction.\" You can see this if you drop a bouncy ball straight to the ground. It will bounce back up, but not quite as high, because of the forces of gravity and friction of the air. So why is it that if you drop something into water, like a sugar cube into a cup of tea, the water will splash back higher than where you dropped the object from?

Thanks!
What do you think?

[JP]

Even in the bouncing ball case, Newton's third law doesn't give the whole answer.  If I drop a rubber ball and a brick onto the floor, the ball will bounce, while the brick won't.  The equal and opposite reaction is enough to stop the ball and brick from moving.  The ball gets an extra "kick" because it compresses when it first strikes and then springs back to its original shape, pushing off the ground with additional force.

You can do a basic analysis of this situation with the law of conservation of energy.  Without going into details, the height something can reach when gravity is pulling down on it depends on the energy it has divided by its mass.  Since the mass of the tea drops is much much smaller than the mass of the sugar cubes, the total height they can reach is higher than the sugar cube's starting point.

Here's the more detailed analysis:
If you hold something off the ground, it has potential energy, which is given by U=m*g*h, where m is the mass of the object, g is a constant that specifies the strength of earth's gravity, and h is the height you've lifted the object.  When it falls, this energy changes into kinetic energy, which is the energy of moving objects.  When it bounces, it rises and slows down: the kinetic energy becomes potential energy again.  At its peak, it can not have gained energy (energy is conserved), so it can't get any higher than it was when you dropped it.  This explains why a bouncy ball never bounces higher than you dropped it.

Ok: so what about a sugar cube in water?  Well the sugar cube has some potential energy due to gravity that depends on its height and mass.  When it hits the water, it has converted that potential energy into kinetic energy.  However, when it enters the water it slows down, so it loses energy.  This energy lost by the cube is transfered into the water: some of it as heat, some of it as waves, and some of it will kick droplets back out of the water.  Remember that the potential energy of something is equal to m*g*h.  Even if they're not getting all the energy from the cube, if these water droplets have a very tiny mass, then you can multiply them by a large height, h, and still have a small total energy for the droplet.  Put another way, if you have a water droplet kicked out of the tea that has energy E, the height it will reach (assuming it splashes straight up) is h=E/(m*g).  So for a fixed energy, if you take a very small mass of the water, it can reach a great height.

[another_someone]

Or put is another way - a big lump of sugar can send a small droplet of water very high, but a small lump of sugar will not be able to move a large body of water very far.

Incidentally, you can get a rubber ball to bounce higher than you drop it - if you give the ball spin, that spin can contain sufficient energy to cause the ball to bounce above its starting position.

[graham.d]

Yes, while those answers are right it is still an interesting situation not wholly answered by conservation of energy or momentum. If you consider a snooker table with unequal weight balls and then see what happens when a larger cue ball impacts a smaller object ball, you will note that the smaller ball does not end up going at a larger velocity than the initial velocity of the cue ball. Ignoring any additioanl spin, both balls will continue moving in the same direction (assuming a full-on impact) with the smaller ball moving, but slower than the original cue ball, and the larger ball slowed down somewhat. The smaller ball has not attained any greater velocity than the cue ball.

The reaction of the fluid is different to this elastic condition in that it slows the sugar cube down rapidly. It is the displacement of the fluid followed by the returning inward backflow that results in the splash upwards. It is not at all obvious that this fluid should be capable of achieving a greater height than the initial height from which the cube was dropped. Having sufficient energy is only a partial answer; the fluid dynamics that is responsible is a much more complex phenomenon. You can get this effect with a meteorite strike when it causes a crater too.

[syhprum]

This phenomena can also be demonstrated with rubber balls, if you drop a small rubber ball and a large one in contact upon reaching the ground some of the energy that go's into compressing the large ball will be transfered to the small one propelling it to a much greater height

[JuliaKia]

Thanks for the great replies to this! Even a moment after sending in the question, I had other examples and permutations to ask about, and you've just given me more! (By the way, the billiard balls are a better example than the rubber bouncy ball of what I was trying to ask about, so thanks for that particular example! Same kind of concept comes across with those desk toys with the five silver balls)

SO, for my new questions - how much does mass have to do with it? If a very massive object bumps into a small (and low mass) object in outer space, does the smaller object move away at the same speed, or a greater speed? Does the bigger object keep moving? If both objects are moving after collision, is it possible to add the speeds of the two together and get the original speed? (I suspect not, but I can't explain why!)

Also, how does this work with fluids in outer space? I think there are some videos of astronauts on the space station sipping airborne droplets of soda with straws. If you were able to get a teacup-sized blob of tea to float in a fairly stable way, and then you launched a sugar lump at it, what would happen? Would the tea move away (although broken up a bit)? Would it "splash" back towards you? What would happen if you just poked it gently with your finger? Or slapped at it with a flat hand?

Sorry the LONG list of questions, but my imagination is running wild!

[LeeE]

Actually, I don't think that snooker/billiard table in the earlier example had been levelled properly:)

The reason why, in graham.d's query where a large ball is sent into a smaller ball and the small ball does not then move off faster than the initial speed of the large ball, is that the energy initially imparted into the larger ball is shared with the smaller stationary ball when the collision happens and they both end up moving.  Because the total energy is the same (allowing for friction losses etc.) but the total mass is higher, the individual speeds will be lower.

In fact, it is possible to play a 'stun' shot where the cue ball (the one imparted with the energy) imparts all of it's energy into the target ball, leaving itself stationary and sending target ball off at the speed of the cue ball (once again allowing for losses).  This is what happens with the Newton's Cradle type toy, with the line of swinging balls suspended on strings.  The key to this type of collision is that they are rigid-body type collisions where no, or in practice, little energy goes into deformation.  In a perfect rigid-body collision, if all the energy from the larger ball is transferred to the smaller ball, the larger ball will stop and the smaller ball will move off faster than the larger ball was moving.

With the bouncy ball situation you have an elastic collision and energy is used to deform the ball.  Because it's imperfectly elastic, not all of that energy is returned when the ball regains it's shape and some of it is lost as heat.

In both cases, not only do you lose energy to heat and friction: unless you're doing it in a vacuum you'll also lose energy through sound.

I'm not so sure about fluid dynamics but I think that when you drop a lump of sugar into a cup of tea it's more like an elastic collision.  The tea acts like an elastic body, due to it's surface tension and the influence of gravity, but once again it's not perfect and it'll be subject to losses.  Some of the kinetic energy from the sugar lump will end up as heat from moving the molecules of tea around but the surface tension of the tea will help stop the tea from simply turning into a mist of individual droplets that just fly out of the cup, and instead store some of the energy in the form of distortions on the surface of the tea.  Due to gravity, these distortions equate to potential energy and as gravity and surface tension even out the distortions the energy stored in them has to be released, and unless the cup is also elastic, the only way for the energy to go is up.

I think that the size of the droplets and how high they go will depend largely upon the nature of the fluid i.e. it's density and surface tension.

This is actually another demonstration of how weak the force gravity really is.  Just one cup of teas' worth of the bonds and forces that bind the particles making the tea can overpower the force of gravity generated by the entire earth and force droplets of tea back up against that force of gravity.

[graham.d]

Just to elaborate on LeeE's comments - the billiard ball stun shot works because the player strikes the cue ball in such a position (below the centre) so that it causes the ball to impact the object ball without actually rolling towards it, but rather skidding. This means it has no significant angular momentum (due to rolling) when it hits the object ball and simply transfers all its kinetic energy to the object ball. I was ignoring the angular momentum issues in my earlier comments and assuming also that the surface of the table was frictionless. The collision of billiard balls is an elastic collision and I assumed we can ignore energy losses. You cannot execute a stun shot if the balls are of different masses - well expert players can put enough backspin on the ball to have a similar effect but that is not the ideal case we are thinking about.

Taking the balls into gravity free space explains more the transfer of momentum. You can then more easily imagine the same situation but assume that the balls (of equal mass) are heading towards each other at the same speed. When they impact it is easy to see that they will both bounce back in the direction from which they came at the same speed. But note that in another frame of reference, one which moves with one of the balls, this is identical to the first case, where (to you) the object ball is stationary and the other approaching. In this case you will see, as in the original case, the cue ball stop and the object ball move off.

In no case with such collisions can either ball attain a higher speed than the original cue ball, even when the balls are of different masses. This is why your initial question is interesting. It does not violate conservation of energy but it is just that it is a more complex reaction than at first sight. If you have an offset see-saw with a long extension one side of the fulcrum, say twice that on the other side, and you place a ball on this long side. Then you drop a ball of twice the mass of the first ball on the short side, the height it attains will certainly be higher than the height from which the ball was dropped. What is interesting is how this mechanical advantage arises from the cup of tea.

[JuliaKia]

Wow! This is really fascinating reading! Thanks everyone!