Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: David Cooper on 03/01/2020 19:04:28
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The Christmas edition of QI asked a question which they failed to answer. You're in a zero-gravity environment (technically micro-gravity, but we don't need to worry about that too much unless it's a draw), and you have an elephant and a mouse to hand. You have a friend some distance ahead of you and you can either throw the elephant or the mouse at him. Which, if either, would lead to your friend moving more when he catches it.
Let me simplify this a bit though by providing a spring which can be used to do all the initial pushing, so it's now more a case of shooting the mouse or elephant at your friend. The spring will be fully compressed before shooting, and fully extended afterwards. The spring will be considered to have no mass, again to simplify things.
Clearly, the mouse will move faster than the elephant, but also, when the spring pushes between you and the elephant, that will send you moving backwards much more than when you shoot the mouse. Will that make any difference to the speed the friend will end up moving at though?
I should provide weights for the four participants. Let's go for a baby elephant weighing 1000kg. The two humans are both the typical modern amount of 100kg. The mouse is also well fed and weighs 1kg.
(I don't know the answer and I'm unsure as to how to calculate it, so I wouldn't trust any figure that I came up with.)
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I would like to know as well, but there is something about this that is giving me pause.
Conservation of momentum would dictate that, if both objects start with a velocity (and therefore momentum) of zero, then the total momentum of the system must remain zero after the spring is released. This means that the momentum of the person and the elephant (or mouse) must be equal in magnitude and opposite in direction to each other so that they still add up to zero. The kinetic energy should be shared equally between the person and elephant (or mouse).
However, I don't think this would necessarily mean that the kinetic energy transferred is the same in both scenarios. In both cases, we have the same spring compressed by the same amount and therefore it stores the same amount of potential energy. So the upper limit of the kinetic energy transfer is the same in both cases. Would this necessarily mean that all of the spring's potential energy is converted in the kinetic energy of motion of both the person and the elephant/mouse? I don't think it would.
Consider an extreme scenario where the spring is accelerating an atom instead of either an elephant or a mouse. If the spring could convert all of its potential energy into the energy of motion of the person and the atom, with both the person and atom sharing equal amounts of kinetic energy, that would send the atom off at greatly hypersonic (or even relativistic) speed. But a real spring can't do that. A spring can only send the atom flying off as quickly as the spring itself can decompress (far less than relativistic speed). So the atom must have only acquired a tiny percentage of the maximum energy that was in the spring instead of a full 50%. The remaining energy in the spring would be converted into a series of further stretches and compressions, eventually dissipating that as heat energy.
For this reason, I'm inclined to say that pushing the elephant would move you more quickly than pushing a mouse, if only by a tiny bit.
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It comes down to conservation of momentum -
m1v1 = m2v2
where
- m is the mass of the object in kg
- v is its velocity in m/s (the sign is important)
- 1 & 2 are the two objects involved in the interaction.
At takeoff, using a compressed (massless) spring
Elephant: 1000ve = 100vh
So the human travels 10x faster than the elephant (in the opposite direction)
Mouse:vm = 100vh
So the mouse travels 100x faster than the human (in the opposite direction)
Let's assume that the friend also has a massless spring of the same kind, so it is an elastic collision...
Elephant: 1000ve = 100vf
So your friend's velocity changes by 10x as much as the elephant
Mouse:1vm = 100vf
So the mouse's velocity changes by 100x as much as your friend
I have to go now, so I'll leave it for others to complete...
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Answering before reading other replies.
The Christmas edition of QI asked a question which they failed to answer. You're in a zero-gravity environment (technically micro-gravity, but we don't need to worry about that too much unless it's a draw), and you have an elephant and a mouse to hand. You have a friend some distance ahead of you and you can either throw the elephant or the mouse at him. Which, if either, would lead to your friend moving more when he catches it.
Depends what you mean by 'move more'. I can throw any tiny thing at my friend at some low speed and it will impart momentum to the friend (and me for that matter) and we will begin to move apart and never stop. So I am guessing you mean to ask which will cause the friend to move faster, and that depends on how much momentum is transferred, not necessarily on the mass of the thing thrown. Still, the elephant masses far more than I do, so if I throw it, most of the energy imparted goes into me and not into the elephant. It seems the mouse is the best way to do it given an equal energy throw in both cases.
Let me simplify this a bit though by providing a spring which can be used to do all the initial pushing, so it's now more a case of shooting the mouse or elephant at your friend.
That's equal energy.
when the spring pushes between you and the elephant, that will send you moving backwards much more than when you shoot the mouse. Will that make any difference to the speed the friend will end up moving at though?
I would guess it makes a difference, but let's see.
I should provide weights for the four participants. Let's go for a baby elephant weighing 1000kg. The two humans are both the typical modern amount of 100kg. The mouse is also well fed and weighs 1kg.
We call those rats around here.
Lets say the elephant moves 0.1m/sec and you move -1m/sec. The friend will acquire around 0.09 m/sec when the elephant hits him. Energy is 50 joules for you and 5 for the elephant, or 55 total.
Same energy with the rat gives you 10 m/sec for the rat and .1 for you. Your friend will acquire similar speed (~0.099 m/sec) so he moves at only slightly faster than he does if the elephant is thrown at him.
Surprise to me. It seems not to make much difference. Did I made a mistake? It would seem I should move faster if pushing off an elephant than if I make a similar energy push off a rat.
Edit:
Looked at Evan's answer and it was very different, but Evan assumes the collision with the friend will be the same energy as the 'throw' energy, which it isn't, especially in the elephant case.
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A spring can only send the atom flying off as quickly as the spring itself can decompress (far less than relativistic speed). So the atom must have only acquired a tiny percentage of the maximum energy that was in the spring instead of a full 50%. The remaining energy in the spring would be converted into a series of further stretches and compressions, eventually dissipating that as heat energy.
Yes, I was wondering about that too, but by making the spring massless, it should then be free to put all that energy into the atom without retaining any in oscillations of its own, and that also frees it to move as fast as necessary to get the atom up to whatever speed it should with all that energy added to it. In the real universe, as the spring has mass, then we have to keep some of that energy in the spring, so it gets more messy. The heavier the mass of the spring though, the less energy we'll lose to that when using the elephant as a projectile rather than the mouse. If we're pushing the elephant or mouse without the spring, it gets harder again in multiple ways (e.g. with heat being generated, and with the push lasting longer when using the elephant, we could apply more force before it's out of reach).
For this reason, I'm inclined to say that pushing the elephant would move you more quickly than pushing a mouse, if only by a tiny bit.
Pushing the elephant would move you a lot more quickly than pushing the mouse because you're then closer to playing the role of the mouse for the elephant and must move faster than the elephant after the push, but it's the speed of the friend that the original question was about.
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My intuitions about the mass of the projectile making a difference seem to be correct, but the values in the example were chosen so there's no difference.
It seems that best way to get the friend moving is to throw something of comparable mass at him. It's a bell curve sort of, and it happens that a 10x mass (elephant) exactly balances the 1/100 mass (mouse) and imparts pretty much the same momentum to the friend in both cases, but a tiny mass (a gram) or a huge one (spaceship) will impart almost zero speed to the friend, and a medium mass (toss my wife) will send him away the fastest. She has that effect.
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Unfortunately, I only had time to look at two half-interactions before I had to catch a plane, so I didn't have a chance to look at the final velocity of your friend.
- I am now overlooking the beach with an onshore summer breeze, so I can now look at the other responses...
Let's assume that the friend also has a massless spring
If your friend catches the mouse/rat, or grabs onto the elephant, I expect that your friend's final velocity would be slightly lower than if they were equipped with a spring(?)
It seems that best way to get the friend moving is to throw something of comparable mass at him.
Yes, there is a similar principle in electrical transmission lines: if you want maximum power transfer, you match the impedance of the source to the impedance of the destination
- With greatly different impedance, not much of the power gets transferred
- This is like the situation with a 100kg human throwing a 1kg rat or a 1000kg elephant - not much energy gets transferred.
it's now more a case of shooting the mouse or elephant at your friend.
Orbital mechanics is tricky, and elephants in orbit move rather slowly (relative to the thrower).
If your friend is ahead of you in orbit, you actually need to fire the elephant backwards - away from your friend.
- The elephant will fall into a lower (faster) orbit, and catch up to your friend on the next orbit (around 90 minutes later, in low-Earth orbit).
- After the collision, your friend will move into a different elliptical orbit
- Space is big - you may never see your friend again
- but the elephant might be easier to find; maybe they should grab onto the elephant?
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My friend considers himself to be the centre of the universe.
From his perspective, he doesn't move when he catches either animal.
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My intuitions about the mass of the projectile making a difference seem to be correct, but the values in the example were chosen so there's no difference.
I chose the values without knowing the answer - the original description just said mouse and elephant (plus the two humans). I'm still trying to find out how you work out how fast the two objects will move apart for a fixed amount of input energy.
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I'm still trying to find out how you work out how fast the two objects will move apart for a fixed amount of input energy.
If spring efficiency wasn't an issue, you could simply divide the total energy equally between both objects. Assuming the velocity is far below that of light, you can determine the velocity by using a rearranged version of the kinetic energy equation Ek = 0.5mv2. To solve for velocity, the equation would be v = √((2Ek)/m).
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I'm still trying to find out how you work out how fast the two objects will move apart for a fixed amount of input energy.
I started with the elephant which is posited to mass 10x myself. So I chose a speed for myself that was 10x the elephant speed, or 1 and 0.1 m/sec respectively. This is an arbitrary speed but not an arbitrary ratio. Then I computed the energy (1/2mv²) for each. 50 + 5 totals 55 joules, so that's the energy of the spring that would impart those speeds on those two objects.
Then I had to do the same with the mouse, but starting with a known energy this time, and I got 0.104 and 10.4 m/sec for myself and the mouse respectively, a ratio of 100 this time. It still adds up to 55 joules.
If spring efficiency wasn't an issue, you could simply divide the total energy equally between both objects.
You can't do that. The elephant gets a 10th the energy that you do when you throw it. The mouse gets 100x the energy when you throw it. Energy can only be divided equally when an equal mass (the wife) gets thrown.
Momentum gets divided equally. Both total momentum and energy is preserved on the throwing event. The former is preserved upon catching event and the energy is lost to friction (or possibly captured if the friend has a smaller spring).
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You can't do that. The elephant gets a 10th the energy that you do when you throw it. The mouse gets 100x the energy when you throw it. Energy can only be divided equally when an equal mass (the wife) gets thrown.
Momentum gets divided equally. Both total momentum and energy is preserved on the throwing event. The former is preserved upon catching event and the energy is lost to friction (or possibly captured if the friend has a smaller spring).
Thank you for the correction. I recently watched a Youtube video claiming that the energy was split evenly, so that's where I went wrong.
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Interesting side question. I throw the elephant at Dewey and he catches it and throws it back with the same energy. The elephant never gets to me. If we do it with the mouse, I can catch the mouse.
Question is: What's the least massive object (assuming 100 kg people) that can't be tossed back to the original person with an equal energy throw? It's somewhere between the elephant and mouse.
I'd give a hint but I follow a golden rule to not give them. ;)
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Interesting side question. I throw the elephant at Dewey and he catches it and throws it back with the same energy. The elephant never gets to me. If we do it with the mouse, I can catch the mouse.
Question is: What's the least massive object (assuming 100 kg people) that can't be tossed back to the original person with an equal energy throw? It's somewhere between the elephant and mouse.
I'd give a hint but I follow a golden rule to not give them. ;)
probably advantageous considering how often you miss the mark! lol
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don't bother sending a letter, I just delete them! lol.
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I started with the elephant which is posited to mass 10x myself. So I chose a speed for myself that was 10x the elephant speed, or 1 and 0.1 m/sec respectively. This is an arbitrary speed but not an arbitrary ratio. Then I computed the energy (1/2mv²) for each. 50 + 5 totals 55 joules, so that's the energy of the spring that would impart those speeds on those two objects.
Then I had to do the same with the mouse, but starting with a known energy this time, and I got 0.104 and 10.4 m/sec for myself and the mouse respectively, a ratio of 100 this time. It still adds up to 55 joules.
Okay, so if I've got the method right, I should be able to test other values like this (the first and fourth reproducing your values) [and note that I'm using * as a multiplication sign instead of x]:-
100kg human vs. 1kg rodent
--> h = (55/101)*1 (kinetic energy of human); and r = (55/101)*100 (kinetic energy of rodent)
--> 50 * v*v = h; and 0.5 * v*v = r (the 50 and 0.5 being half the weights)
--> v = root(h/50 or r/0.5)
--> v = 0.10436 (speed of human) or 10.436 (speed of rodent).
100kg human vs. 10kg dog
--> h = (55/11)*1 (kinetic energy of human); and d = (55/11)*10 (kinetic energy of dog)
--> 50 * v*v = h; and 5 * v*v = r (the 50 and 5 being half the weights)
--> v = root(h/50 or r/5))
--> v = 0.316227766 (speed of human) or 3.16227766 (speed of dog)
100kg human vs. 100kg human
--> h = 55/2
--> 50 * v*v = h
--> v = root(h/50)
--> v = 0.7416198
100kg human vs. 1000kg elephant
--> h = (55/11)*10 (kinetic energy of human); and e = (55/11)*1 (kinetic energy of elephant)
--> 50 * v*v = h; and 500 * v*v = r (the 50 and 500 being half the weights)
--> v = root(h/50 or e/500)
--> v = 1 (speed of human) or 0.1 (speed of elephant)
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Okay, so if I've got the method right, I should be able to test other values like this (the first and fourth reproducing your values) [and note that I'm using * as a multiplication sign instead of x]:-
100kg human vs. 1kg rodent
--> h = (55/101)*1 (kinetic energy of human); and r = (55/101)*100 (kinetic energy of rodent)
--> 50 * v*v = h; and 0.5 * v*v = r (the 50 and 0.5 being half the weights)
--> v = root(h/50 or r/0.5)
--> v = 0.10436 (speed of human) or 10.436 (speed of rodent).
So far so good. That's the values I got, specified to 3 digits in reply 10.
100kg human vs. 10kg dog
--> h = (55/11)*1 (kinetic energy of human); and d = (55/11)*10 (kinetic energy of dog)
--> 50 * v*v = h; and 5 * v*v = r (the 50 and 5 being half the weights)
--> v = root(h/50 or r/5))
--> v = 0.316227766 (speed of human) or 3.16227766 (speed of dog)
100kg human vs. 100kg human
--> h = 55/2
--> 50 * v*v = h
--> v = root(h/50)
--> v = 0.7416198
100kg human vs. 1000kg elephant
--> h = (55/11)*10 (kinetic energy of human); and e = (55/11)*1 (kinetic energy of elephant)
--> 50 * v*v = h; and 500 * v*v = r (the 50 and 500 being half the weights)
--> v = root(h/50 or e/500)
--> v = 1 (speed of human) or 0.1 (speed of elephant)
Looks good. None of the calculations go so far as to compute how fast the friend ends up moving by catching the thrown thing. It's a little faster for the mouse, but much faster for the middle masses.
What is the optimal mass to get my buddy moving the fastest? Certainly not either the elephant or mouse. I suspect the answer is the same as my prior question: He moves fastest with the throwing of a mass that is just large enough that it can't be thrown back.
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probably advantageous considering how often you miss the mark! lol
Are you making reference to that ridiculous post of yours that got deleted? The one where you claimed that the surface area of the objects matters when it comes to the momentum and kinetic energy transfer?
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How can a 1000 kg elephant moving at 1 m/s hit a standing 100 kg man and makes him move at 10 m/s by transferring all of its momentum?
It can be done if the elephant actively kicking him while they are in contact. A passive elephant can't do that.
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How can a 1000 kg elephant moving at 1 m/s hit a standing 100 kg man and makes him move at 10 m/s by transferring all of its momentum?
It can be done if the elephant actively kicking him while they are in contact. A passive elephant can't do that.
Well, the man can do the kicking as well. It would be nearly the same energy that it took to get the elephant up to that speed in the first place, so clearly the participants have a bigger spring or something.
Are you under the impression that anybody suggested that the thrown object transfers all its momentum to the friend? The OP makes no mention of that. The guy catches the elephant and the two move together thereafter. I suggested (in reply 12) that he might attempt to throw the elephant back to me, but it would take a lot more energy than the original throw to do that.
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We should move gently and care and love with life partner
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How can a 1000 kg elephant moving at 1 m/s hit a standing 100 kg man and makes him move at 10 m/s by transferring all of its momentum?
It can be done if the elephant actively kicking him while they are in contact. A passive elephant can't do that.
Well, the man can do the kicking as well. It would be nearly the same energy that it took to get the elephant up to that speed in the first place, so clearly the participants have a bigger spring or something.
Are you under the impression that anybody suggested that the thrown object transfers all its momentum to the friend? The OP makes no mention of that. The guy catches the elephant and the two move together thereafter. I suggested (in reply 12) that he might attempt to throw the elephant back to me, but it would take a lot more energy than the original throw to do that.
I missed the word catch in the OP. In this case, the objective is maximising the momentum of the friend, which means minimizing the momentum of the projectile after being caught. Since they have the same final velocity, it means minimising the mass of the projectile. So the answer to the question is that he should use the mouse.
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minimizing the momentum of the projectile after...
You can reduce it more than you expect, by using a spring: This bounces the mouse away from your friend, making the rodents momentum negative (in the frame of reference of the friend+mouse).
by transferring all of its momentum?
Perhaps you are familiar with the toy sometimes called "Newton's balls"?
Here, one swinging ball manages to transfer all of its momentum and energy to another ball, the original ball almost halting.
This is only possible because:
- The balls have identical mass
- The balls are steel, and highly elastic
- The balls strike head-on
- Vary any of these conditions, and the velocity transferred will be reduced
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You can reduce it more than you expect, by using a spring: This bounces the mouse away
This violates the OP conditions that the friend catches the projectile, not using it as reaction mass for further thrust in the direction of his choice. If the latter, I suppose the best solution is to throw the largest thing possible, have the friend use nail clippers to cut it into tiny pieces and launch each piece with the spring, thus implementing a crude rocket engine.
If we allow the original projectile to bounce off intact then it changes the answer, but a middle-size projectile still gets the best results. You get 2x the momentum transfer compared to catching it, which ranges from nearly 0% for large objects to 200% for the smallest objects, but 200% of negligible is still negligible, so some figure in between is optimal for getting your friend moving the fastest.
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I can't remember the exact wording of the question on the TV programme where it came from: it may not have stated that the mouse or elephant is caught, so it could bounce instead. If we put a spring there too with no energy stored in it, the projectile will then compress the spring, leading to the projectile plus target moving the same speed for a moment, giving us one result, and then the spring would release that energy again to give us an elastic collision, giving us the other result.
For example, with a human projectile of 100kg, the projectile would accelerate the target until their speeds match for a moment, and then the spring would halt the projectile and the target would then move at the speed the projectile travelled at before it hit the target. That's Evan's Newton's balls thing. (They kept Einstein's brain too, but that's another story.)
Anyway, I'm now trying to work out how to calculate the two cases for the full variety of masses: the combined speed of projectile and target when their speeds match for a moment, and their speeds after the elastic bounce. (There's no need to explore what happens if a new kick or push is applied as that's stored energy being released which didn't come from the original spring's push.)
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On a related note, does a film of a man throwing a mouse look the same as a film of a man catching a mouse, played backwards?
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does a film of a man throwing a mouse look the same as a film of a man catching a mouse, played backwards?
Since it is hard to throw a mouse through your center of gravity, the throw ends of with the human tumbling and/or spinning.
It is possible to catch a mouse at your center of gravity (but highly unlikely), so the catch probably ends with the human tumbling and/or spinning.
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This looks like the right magic spell to chant for the end result of the elastic collisions: bottom of page https://www.real-world-physics-problems.com/elastic-collision.html (https://www.real-world-physics-problems.com/elastic-collision.html), but we only need the left-hand part as the initial speed of the target is zero.
So, final speed of projectile = ((projectile mass - target mass) / total mass) * initial projectile speed
and final speed of target = (2 * projectile mass / total mass) * initial projectile speed
100kg human vs. 1kg rodent
--> h = (55/101)*1 (kinetic energy of human); and r = (55/101)*100 (kinetic energy of rodent)
--> 50 * v*v = h; and 0.5 * v*v = r (the 50 and 0.5 being half the weights)
--> v = root(h/50 or r/0.5)
--> v = 0.10436 (speed of human) or 10.436 (speed of rodent)
So, we now have ((1-100)/101) * 10.436 = -10.230 for the rodent's final speed
and ((2*1)/101) * 10.436 = 0.2067 for the human target's final speed
(The mouse will return to the first human and hasn't lost a lot of speed, so we could have a series of further bounces to accelerate both humans to higher speeds.)
100kg human vs. 10kg dog
--> h = (55/11)*1 (kinetic energy of human); and d = (55/11)*10 (kinetic energy of dog)
--> 50 * v*v = h; and 5 * v*v = r (the 50 and 5 being half the weights)
--> v = root(h/50 or r/5))
--> v = 0.316227766 (speed of human) or 3.16227766 (speed of dog)
((10-100)/110) * 3.1623 = -2.587 for the dog's final speed
(2*10/110) * 3.1623 = 0.5750 for the human target's final speed
(The dog will not be able to catch the first human, so no further bounces will occur.)
[Edit: thanks Halc for pointing out that that's wrong and that the dog will catch the human easily - we can have a series of further bounces with the dog too.]
100kg human vs. 100kg human
--> h = 55/2
--> 50 * v*v = h
--> v = root(h/50)
--> v = 0.7416198
((100-100)/200) * 0.7416 = 0 for the human projectile's final speed
(2*100/200) * 0.7416 = 0.7416 for the human target's final speed
100kg human vs. 1000kg elephant
--> h = (55/11)*10 (kinetic energy of human); and e = (55/11)*1 (kinetic energy of elephant)
--> 50 * v*v = h; and 500 * v*v = r (the 50 and 500 being half the weights)
--> v = root(h/50 or e/500)
--> v = 1 (speed of human) or 0.1 (speed of elephant)
((1000-100)/1100) * 0.1 = 0.08181 for the elephant's final speed
(2*1000/1100) * 0.1 = 0.1818 for the human target's final speed
So, with elastic collisions, it's better to throw the 1kg rodent than the 1000kg elephant, and much better still if you include the effect of further rodent rebounds.
I still don't have results for collisions where the projectile stays with the target afterwards (in which cases some energy will be left stored in the spring between them).
One of the things I wanted to understand from this is how it can be that you can send more kinetic energy to someone using the rodent and yet end up with the target moving more slowly than if you send a person to them carrying less kinetic energy. The answer to that puzzle is that the rodent retains most of that extra kinetic energy by bouncing back, and to make sense of what happens before it bounces back, it compresses the spring more than in the human-projectile case, so that's where all the extra energy is hidden, and when it's released, most of it goes back into the rodent instead of the target. It all makes sense now.
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and final speed of target = (2 * projectile mass / total mass) * initial projectile speed
This is if it bounces off (elastic collision). Get rid of the 2* if the target catches the projectile. The latter is the original query.
The mouse will return to the first human and hasn't lost a lot of speed, so we could have a series of further bounces to accelerate both humans to higher speeds.
That it will. What is the cutoff mass for this to happen? This is not the same mass as the answer to my post 12 question.
100kg human vs. 10kg dog
...
((10-100)/110) * 3.1623 = -2.587 for the dog's final speed
(2*10/110) * 3.1623 = 0.5750 for the human target's final speed
(The dog will not be able to catch the first human, so no further bounces will occur.
Oops. Dog is moving at -2.587 and the first human at -0.316. It catches up easily.
((100-100)/200) * 0.7416 = 0 for the human projectile's final speed
Just like the balls in the picture in post 22.
(2*1000/1100) * 0.1 = 0.1818 for the human target's final speed
So, with elastic collisions, it's better to throw the 1kg rodent than the 1000kg elephant, and much better still if you include the effect of further rodent rebounds.
Given your 1 vs 1000 kg masses yes. Realistic figures are more like 25g vs 6000 KG. Still a close race, and the target human moves dang slow in both cases. I get the friend moving at ~0.016 m/sec after catching (not bouncing) either mouse or elephant. Ouch. Need more significant digits.
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I can't remember the exact wording of the question on the TV programme where it came from: it may not have stated that the mouse or elephant is caught, so it could bounce instead. If we put a spring there too with no energy stored in it, the projectile will then compress the spring, leading to the projectile plus target moving the same speed for a moment, giving us one result, and then the spring would release that energy again to give us an elastic collision, giving us the other result.
I didn't realize sooner that the word catch implies that the projectile and the friend would have the same speed afterward. Here are some definitions related to this thread.
1. intercept and hold (something which has been thrown, propelled, or dropped).
"she threw the bottle into the air and caught it again"
and an alternative definition
6. strike (someone) on a part of the body.
"Ben caught him on the chin with an uppercut"
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Since it is hard to throw a mouse through your center of gravity, the throw ends of with the human tumbling and/or spinning.
It is possible to catch a mouse at your center of gravity (but highly unlikely), so the catch probably ends with the human tumbling and/or spinning.
It is possible if you throw using both arms like in the picture below, and you pose like flying superman by straightening your body and legs, aligned with projectile's trajectory. This shouldn't be too hard in microgravity environment.
(https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQrRKcLsjI5x6DmxO3dp0sveviGQqFHJgcj28ABBbiDVGDsvRh6&s)
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minimizing the momentum of the projectile after...
You can reduce it more than you expect, by using a spring: This bounces the mouse away from your friend, making the rodents momentum negative (in the frame of reference of the friend+mouse).
by transferring all of its momentum?
Perhaps you are familiar with the toy sometimes called "Newton's balls"?
Here, one swinging ball manages to transfer all of its momentum and energy to another ball, the original ball almost halting.
This is only possible because:
- The balls have identical mass
- The balls are steel, and highly elastic
- The balls strike head-on
- Vary any of these conditions, and the velocity transferred will be reduced
I wonder if in the configuration of 3 vs 2 balls as shown at 0:22 in the video, the balls are glued so they can't be separated by the collision. Is it possible that the collision still elastic?
If 3 glued ball start with velocity -1 m/s (moving left), what are the speed of 3 balls and 2 balls after collision?
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I agree kryptid. And if it isn't a spring but a human trying to 'push' that elephant in a 'flat space' the most probable thing would be the human moving, not the elephant.
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I know, action and reaction, the elephant will 'move' too, to a degree.
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Get rid of the 2* if the target catches the projectile. The latter is the original query.
Thanks - I was still wondering how to do that.
The mouse will return to the first human and hasn't lost a lot of speed, so we could have a series of further bounces to accelerate both humans to higher speeds.
That it will. What is the cutoff mass for this to happen? This is not the same mass as the answer to my post 12 question.
I don't know how to calculate that without using trial and error. You likely use calculus, but I'd write a program to try lots of values, if only I had the time. It occurs to me though that if a mouse bounce was to leave the mouse stationary at any point, I suspect that would leave the two humans moving apart at the same speed they would have done if they'd just been pushed apart directly by the original spring.
Oops. Dog is moving at -2.587 and the first human at -0.316. It catches up easily.
Thanks for spotting that. I might never have noticed that I misread the place of the decimal point.
For completion then, the inelastic collision results are half the elastic ones:-
(1/101) * 10.436 = 0.1033 for the human target + rodent's final speed [rather than 0.2067]
(10/110) * 3.1623 = 0.2875 for the human target + dog's final speed [rather than 0.5750]
(100/200) * 0.7416 = 0.3708 for the human target + human projectile's final speed [rather than 0.7416]
(1000/1100) * 0.1 = 0.0909 for the human target + elelphant's final speed [rather than 0.1818]
I didn't expect it to be that simple, but if you change frame of reference to travel with the moving pair of animals before they ping apart again, you can see that you get the same speed gain for the target human again at the end rather than the reduced push that the projectile effectively receives.
One more thing on a different but related issue. If I wanted to write a snooker simulation but with variable-mass balls and with the angles of impacts leading to the balls going off at different angles, can the same maths as above still be used just by breaking things down into vectors? The path the target ball follows is easy to predict, so that direction and the one perpendicular to it can be used as vectors for the cue ball, leading to one of those vectors surviving the impact for the cue ball unchanged, while the other vector would adjust exactly as in the straight-line case. Does that sound viable or do you think it would be more complicated?
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What is the cutoff mass for [projectile returning to first person]? This is not the same mass as the answer to my post 12 question.
I don't know how to calculate that without using trial and error. You likely use calculus
It's actually just complicated algebra, but I did my reply-12 thing by guessing, and hit it in my 2nd guess, the first guess being 50kg. I had a hunch and it turned out to be correct. This is the case of the 2nd guy returning the projectile with the same energy as the first throw, not at the energy of the incoming projectile.
One more thing on a different but related issue. If I wanted to write a snooker simulation but with variable-mass balls and with the angles of impacts leading to the balls going off at different angles, can the same maths as above still be used just by breaking things down into vectors?
Snooker is complicated. It isn't frictionless balls in a vacuum, which is actually pretty easy arithmetic. Snooker has to deal with table friction, especially immediately after the impacts where the balls are moving at a different rate than their spin. That spin deflects both balls from their initial trajectories, and I'm not up to the challenge of computing the angular inertias and funny spins with non-horizontal axes.
Yes, it's quite easy to do with frictionless balls in space. Target ball moves directly away from its impact point and incoming ball is deflected in a way that preserves momentum and energy.
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Are the mouse and the elephant wearing spacesuits?
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Are the mouse and the elephant wearing spacesuits?
Given the weight I gave for the mouse, I expect it is.