Bouncing fruit collision example

there's a big fat juicy apple hanging from a tree branch and you want this Apple but you can't climb the tree luckily you've got an orange in your pocket so you take this orange and you chuck it at the Apple and it strikes it at the apex of its trajectory causing the Apple to fly off and now you've got an orange and an apple now this is technically fruit vandalism if it's not your apple tree so make sure you're only picking your own apples or you've paid someone to do this and everything is legit but this is also a collision problem and in physics you could solve for the velocities involved in the mass and the momentum is involved by using conservation of momentum if we had some numbers so let's give ourselves some numbers let's see if we can solve for some quantities here so let's say this Apple I told you is big and fat let's say this Apple was 0.7 kilograms let's say the orange is probably not as big so 0.4 kilograms and let's give some numbers let's say the speed of the orange right before the collision was five meters per second so since this orange was at its apex right it was just heading in this way and it was going horizontally at that moment five meters per second right before it struck the Apple and let's say the Apple was moving three meters per second right after the collision so right after the orange hit the Apple the Apple starts flying at 3 meters per second one question we could ask one obvious question is well if this is the speed of the Apple after the collision what was the speed of the orange after the collision what was the velocity of the orange and which way was it going so we'll call it V o for V orange and was that orange going left or right immediately after the collision took place sometimes this isn't so obvious so let's see if we can solve for this now we've got enough to solve we can do this using conservation of momentum and conservation of momentum says that if there's no external impulse on a system and our system here is the orange and Apple if there's no external impulse on these fruit that means the total momentum before the collision took place so right before the collision took place has got to equal the total momentum right after the collision took place and is import that we denote right before and right after we're not talking like right as someone through this fruit before it got up here and we're not talking finally like after the Apple gets back down to the ground he can't do that for most collision problems you're going to want to consider right before the collision and right after and the reason is remember this formula here is only true if there's no external impulse so only if external impulse is zero and you might be like whoa isn't it always zero as it shouldn't it be zero in this case it's not so obvious if you're clever you might be like wait a minute there's a force of gravity on this Apple there's a force of gravity on the orange so doesn't that mean there's an external force and if there's an external force doesn't that mean there's an external impulse and does that mean that the momentum shouldn't be conserved well not really the reason is for one this force of gravity is directed downward so it's only going to affect the vertical momentum and we're just talking about the horizontal momentum here I want to know what happens to the horizontal momentum of this orange but secondly the definition of impulse is that it's the force that X multiplied by the time duration and we're going to say that if we consider right before the collision and right after the collision is our initial and final points this time interval is going to be so small that the force of gravity is going to have almost no time to act and because it has almost no time to act it has almost no external impulse so we're going to ignore the impulse due to gravity because it acts over such a small period of time and it's such a modest force which means we get to use conservation of momentum for our system so was this going to look like well the momentum formula is mass times velocity so the initial momentum of the system let's see I'd have to add up initial momentum of the orange is 0.4 kilograms that's the mass times the initial velocity that's five meters per second plus I'm going to add to that the mass of the Apple 0.7 kilograms multiplied by the initial velocity the Apple what was the initial velocity the Apple it wasn't three people try to plug in three that was the final velocity Apple the initial velocity the Apple was just zero because it was hanging on a tree branch and was just sitting there so there's going to be zero what that means is this entire term is going to be zero because zero times seven zero point seven is still zero this term just goes away it's going to be zero equals the final momentum RH so we added up the total momentum of our system initially now we're going to add up all the momentum of our system finally so zero point four kilograms it's the mass of the orange multiplied by we don't know the final velocity Orange that's the thing we want to find so I'm going to write that as V o4v of the orange this final velocity of the orange is what we want to find this term here represents the final momentum of the orange but I have to I can't stop you I have to add to that the final momentum of the apple so remember when you're writing down conservation of momentum for a system the statement isn't that the initial momentum of one object equals the final momentum of some other object it says that the total initial momentum of the entire system equals the total final momentum of the entire system so I'll take my point seven multiplied by my final velocity is three meters per second for the Apple and now I can solve so we can solve this I've only got one unknown so 0.4 times 5 is 2 kilogram meters per second plus zero so I'm not going to write that because if just take up space equals 0.4 times V PO is the unknown so 0.4 kilograms times the unknown vo and then plus 0.7 times 3 is going to be two point one kilogram meters per second so our system started off with two kilogram meters per second of momentum to the right that's what the orange brought in and our system ends with 2.1 kilogram meters per second to the right which is what the apple has plus whatever momentum the orange has right after the collision and you might look at this and be like wait a minute oh we screwed something up two kilogram meters per second equals two point one kilogram meters per second plus something how's this right-hand side ever going to equal to if it's got two point one two star with but remember it can momentum is a vector and vectors can be positive or negative depending on whether they point right or left so this just tells us okay the orange is going to have to have momentum leftward after the collision so that this whole right-hand side can add up to two again and we know we're going to have a final velocity of the orange that's negative but you don't have to be clever like if you just wanted to solve this equation it'll tell you whether it's going right or left I'll show you why if we just if we just do this two minus two point one so if we subtract two point one from both sides we'll get negative zero point one kilogram meters per second and that's going to equal this final momentum of the orange so equals zero point four kilograms for the orange times V o the final velocity of the orange and now if we just divide both sides by 0.4 we'll get negative 0.25 meters per second that's the final velocity of the orange and you realize oh I didn't have to figure out the sign beforehand I can just solve and the conservation of momentum formula will tell me whether it's going right or left because if I get a negative sign here just says that velocity had to be directed in the negative direction in order to conserve momentum in this case so this orange right after the collision was heading leftward that's what the negative sign means and this point two-five means it was heading leftward at a rate of 0.25 meters per second so recapping we could use conservation of momentum to solve for an unknown velocity by setting the total initial momentum of the system equal to the total final momentum of the system we got to be careful with negative signs if there was an initial velocity that was negative we would have had to plug in that velocity with a negative number and if we find a negative velocity to end with that means that that quantity that velocity was directed in the negative direction also we can only use conservation of momentum whenever the external impulse is zero which is why we consider points immediately before the collision and immediately after so that this time interval is so small gravity can't apply much of an impulse at all and I should say we should assume that this down was barely hanging on by a string because if this stem was secured to the tree then there would have been an external force that could have caused an external impulse so let's assume this Apple was already just about to fall off and the slightest of forces could knock it off that way there's no external impulse and we get to use conservation of momentum