okay class let's go here um we're today going to be doing the final portion of our kinetics lecture I know it's been a lot of uh lecturing but uh this is the final part and we're going to be going over impulse and momentum uh we this will come up a lot I think during uh we have a lab on this and there'll be some lectures on this as well so or some U exam questions on this as well so make sure pay attention uh most of you guys probably know uh the concept of momentum but maybe not as much on the concept of what impulse is so um yeah let's get into it so uh first thing I want to do is look at Newton's Second Law here uh we I mean we can revisit this I mean we've talked about this a lot it's going to equal for equals mass time acceleration the biggest issue with Newton's second law is that it's just a snapshot of the force acting on an object and the resulting acceleration at that very moment so we know in reality we typically don't add Force at the same amount over a period of time like force will fluctuate you know over a period of time like think about if you're jumping you're not going to be applying the same amount of force to the ground in order to get yourself off the ground consistently throughout you're going to have points where you're having higher or lower or Peak Force so uh it's going to we need to kind of revisit how we look at this equation in order to maybe make it better so we can look at the forces over a period of time and this is where you know the impulse and momentum equation will come into play and we'll go over exactly that on this next slide so impulse momentum equation is derived from Newton's Second Law which is f equals mass times acceleration okay so we're more concerned with how we can change this using algebra into a new equation so first things first let's look at it I mean if we're talking about Force equals mass time acceleration we can redo this using algebra and to have it equal acceleration equals uh change in velocity over time and again the way we did this was basically just reworking what force equal what Ma with acceleration equal okay um we can do this again using these equations right here so Force equals mass time velocity over time how does this work well acceleration is equal to velocity over time so what we're doing is we're just subbing this a for velocity over time and you can even further use some algebra in order to get it to equal force uh time time equals mass time velocity or this equation right here which is the impulse momentum equation which is basically Force time time equals mass time change in velocity velocity final minus velocity initial so that's kind of how we can break this equation down into a new one just using algebra that might better fit we're going to be looking at and this equation is better for looking at change in Vel velocity during different activities like running jumping or even lifting weights so it's better at getting us more of less of a snapshot at one Peri period of time just by reworking the equation okay so so we have this impulse Force impulse Force time time and momentum ma um we have this equation so when we're talking about linear momentum this is essentially the product of an object's mass and its linear velocity we're talking about linear momentum hence the mass times velocity equation that creates linear momentum uh which we can see uh down here in a second um Force um Force equals time oh hold up sorry about that things were l in on my end um so again as we were saying momentum can be defined as l or P equals mass times velocity where this p is the momentum mass is the mass and velocity is velocity or we can put this as L as well so down here so what this means is that any object has momentum that has momentum and the faster the object is or the heavier the object is the greater momentum it's going to have so it we increase this mass or this velocity we're going to increase this momentum here so think about a bowling ball versus a basketball rolling it at the same speed because the bowling ball has a greater mass it has momentum or its momentum is higher okay so it's going to be harder for it to stop it's going to hit the pins and knock all of them down a lot a lot more is going to have a lot more force behind it um next we have the impulse momentum relationship which is a more practical way of interpreting this Newton Second Law which we were talking about earlier when we were trying to figure it out instead of just thinking of forces at a specific moment of time we can actually concern ourselves of how forces act over a period of time and how that change acts and changes an object's momentum so the a average net force is what we have here average net force over a period of time interval which is usually given to us will change an object's momentum which is this mass times change in velocity which we can consider math times velocity is momentum so we see a change in an objects momentum so the average and that Force applied over time we'll change this um and and again we have this equation right here that kind of shows that relationship so what this tells us is that the change in momentum which is mass times final velocity minus initial velocity that's you know whatever your starting momentum is versus your final momentum is direct directly related to an Impulse generated impulse which we'll get into in a second is going to going to equal force over a period of time this is what impulse is we'll get into what that means exactly so it's directly related to the impulse so if we know the forces and how long they're applied we can predict how much an object's momentum is going to change this principle is essential for understanding real world motion like sprinting kicking a ball okay um when we increase momentum using this equation so we're increasing this initial velocity to final velocity so the momentum is going to go up um we I mean we can just see this in a lot of different sports tasks um it think about you know speeding up during a race uh or hitting a baseball with a bat the higher the force you hit it with or the longer you hit that force or that object with or you're applying that Force to an object with the more momentum it's going to have at the end the more you're going to increase that change in momentum um and again that's explained through this property here uh again caused by a large average force the higher the more we add Force into it the over a longer period of time the more it's going to do that's why you want to swing through with a a baseball bat part of the reason is you're going to have more contact time with the ball the longer you apply that Force the ball the farther it's going to go the more momentum it's going to have on the other end of it okay and we can think about this from the decreasing end of it too so often we'll have situations where we want to reduce the speed of an object taking it from a fast initial velocity so something High here to a low over here okay uh this could be anything from catching a ball slowing down in a car um you think about this you know the one example I really want to cover is jumping off a chair uh or jumping off an object at any you know any object uh you know the higher up you are according to our propel a propulsion lecture uh the more speed you're going to gain when hitting the ground because you have more time to increase this velocity here on this end well obviously we want to decrease that speed and stop it or else you're just going to keep falling through falling to the ground which is impossible you're gon you're going to stop when you hit the ground so we're going to see this go to Zero from a high speed maybe 10 to zero when we see that change we're going to see it larger impulse if the timing aspect is bigger so this impulse is going to be the same but what we can change here is the timing aspect of it by like bending your knees increasing the length of how long it's going to take for you to come to a complete stop that's going to decrease the amount of force being applied to your legs so if we increase this timing aspect we can actually decrease this okay and that's important because think about if you're jumping off an object and you just have your legs completely straight out all that force is going to be applied him a small amount of time so you're going to have a really big impulse that big impulse is going to cause damage you know that's why people can like you know break their ankles when they're falling um if you another example would be like a pole vault you know how they have a big cushion on the end other end of the pit that cushion actually slows down your fall so once you hit it you're dispersing the forces over a longer period of time into that cushion instead of falling onto say the ground that cushion is actually increasing the amount of time it's taking for you to slow down your momentum same thing when you're like slowing down a car if you slam on the breaks you're going to feel a lot of impulse big change of momentum is going to you're going to feel that quite a bit but if you slow down gradually you increase the timing aspect of it then you won't feel it as much you won't have a greater impulse NE greater impulse can cause damage um so another little bit of review again remember we talked about Statics Statics is basically net forces equals zero because we don't have acceleration we're not changing acceleration or velocity um Dynamics on the other hand is going to be having an acceleration and you're changing either you know in horizontal vertical Med lateral or all of the directions so you're going to see a net force being applied so again we have these changes in forces now when we're going back to this momentum aspect of these changing forces we know that Newton's first law states that an object will remain in motion unless acted upon by an outside Source okay so that's this first law of you know it's a Newton's first flaw I don't know why I was trying to make it more complicated um so again if there's a body segment that's moving at a constant velocity that segment is going to continue moving at the same velocity unless acted upon by an external Force where do those forces come from it could be a muscle could be an object it could be you know if you're throwing a punch it could be the punching bag that you're hitting so something outside of its direction of travel is going going to stop it from traveling in that direction or change its direction by changing its directional direction of velocity so as it relates to momentum a body segment and its quantity of motion we can look at how much that segment segment is moving and we can Define that its momentum by it what it the momentum is by quantifying it numerically okay in this case we're considering momentum as p uh I kind of flip-flops between L and P I've seen it both ways so um I prefer L but um so let's just say you know in this situation Newton's first law momentum of a body is conserved unless acted upon by another object so um all the momentum is going to stay Within the equation meaning it's not going to go anywhere else whatever we bring to the equation is going to stay in the equation unless something else is acting to stop it or change that so we look at a conser conservation of linear momentum aspect so that's kind of leading us into this so as we know sum of all forces equals mass times acceleration which is let a little bit well that's fine oh um my end or not so again what I was kind of getting out here is the basic principle is when there's no external forces acting upon an object or a system is going to have a constant momentum so principle is really useful when understanding collisions impacts and any situation where forces are involved over a period of time so we can start with Newton's Second Law Force equals mass times acceleration as we know this tells us that the force acting on an object is equal to its mass times acceleration um but acceleration can also be described in terms of change in velocity over time so as we can rewrite this formula here as we kind of did before we could change it to apply more to the momentum aspect so we can change this to be change in velocity over time we know if we take out the changing aspect of it is change in mass times velocity was mass times velocity well mass times velocity is momentum so if we see a change in momentum over time that's going to equal a net force or net change in force or some type of force so this total momentum remains constant with no external Force being applied this principle is called the conservation of linear motion which tells us that in a system where no external forces are present the momentum before and after an event like a collision will remain the same so this is important in the real world like with sports and car crashes or any activity where Force interacts forces are interacting with each other so because it helps us predict the outcome based on how much momentum is being transferred from one object maybe to the other and this is kind of like the final version of this formula I want to say show you here change in P which is momentum okay I know what we've done it both ways with L but change in P over time so if net forces are equal to zero and net forces equal mass times velocity we can say that change of momentum is going to equal zero because the amount of forces acting on it is going to be zero so let's look at this example so we have two footballs here football players here player A and B colliding on a headon like tackle or whatever this is going to be like they're just colliding we know that the previous slide that momentum is conserved in a system with no external forces acting on it this means the total momentum before the Collision is equal to the total momentum after the Collision so here's the basic formula of the momentum conservation so we have this m a * VA plus M MB * VB equals ma plus MV times velocity final which is just a v A+ B okay so um in this situation we can kind of look at a couple different variables here where mass of person a is equal to 90 kg velocity of person a is 2.4 meters per second person B is 110 kilograms and he has a velocity um or actually we don't yeah velocity U before the Collision so his initial velocity is -3.5 so note the negative sign here because the player is moving in the opposite direction of a so typically to the left um we can calculate the momentum of each player before the collisions so player A's momentum is going to equal 260 kg uh kilogram uh time meters per second um so that's say momentum that's the units is kg * m/ s which makes sense because you're just adding kg to Velocity um player B is going to be opposite 385 so negative 385 so again it's going to be a negative momentum because he's going to the left the opposite direction so we can look at total momentum before collision and we can add these together into a system oh why is it going oh sorry this is kind of the example I was just referring to here's the equation here are the variables we were given this is precision so again the total systems momentum remains unchanged so the system is not losing anything so this final is going to equal this initial this final uh Mass final velocity is going to equal the combination of these two right here so we add these two impulses together it's going to equal 169 and now this is kind of confusing but we're actually going to add their masses together CU now they're think of them as a system together so now they're both moving in the same direction so once they Collide they become the same system so we add these two weights together as well and also you know as you can see here the person on the right had more momentum so he's the one that won out so now we're going to the left so we have a momentum going to the left and we can see here by you know doing a quick uh adding here to get 200 dividing doing some algebra we get a final velocity of 84 5 meters per second going to the left so we're now after this Collision occurred the right side went out he had more momentum and now we're rolling this way okay so again note I mean this person a was bigger than person B but the person B had a higher speed which out made the momentum go up larger than the person a okay let we do this simple addition right here so momentum is conserved so we have the same momentum starting and ending okay so we have we had a positive and a negative momentum we add those together we still have a negative momentum because we had more so now we're just moving to the left okay okay so let's move on to changing momentum so when momentum is not conserva so while we've been talking again about scenarios where momentum is conserved with the football players it's important to understand that momentum is not always conserved particularly when external forces are kind of coming into play um so Newton's Second Law we're going to restate this when an O when an external Force acts upon an object over a period of time it will cause a change of momentum the change of momentum is proportional to the net external forces applied to the body so this can be expressed by this equation right here change in forces equals change in momentum over time so again change of forces is the sum of the external forces acting on the body or the object change in P is the change of momentum which represents just how much momentum is changing what the velocity is changing and then over time is time is inverse so meaning over a period of time we're going to see how fast or slow something occurs so the key takeaway here is that the net external for force is not zero momentum will change this is important to think about in real world scenarios like for example if you're driving a car and you hit your brakes the external forces like friction and air resistance resistance will start to act on the car slowing it down which is a clear change in its momentum it's going to lose momentum so where does the M the momentum not conserve where does momentum not get conserved momentum might not be conserved in situations like a soccer ball being kicked a car coming to a stop or friction acts as an external force on the tires or um yeah car company to stop is the best one I like to talk about but in each of these situations the net external force acting on a system like the foot and a soccer ball the friction of the tires means momentum changes and it's not conserved so in summary if there's an external Force being applied momentum will change over time and that change will depend on how big the force is that's acting along it and how long it's being applied for so let's take an example here um we can start with the kicking of a soccer ball when a player kicks a ball they have by a force through their foot for a short you know amount of time it's not going to be in contact for that long this going to generate an Impulse on the ball this Ball's change of momentum is directly related to the force applied and the time the foot remains in contact with the ball so the harder and longer the kick is the greater the change in velocity of the ball is so that's why I was talking about you know you follow through it gives you more contact time with the object so if you just do a Chip Shot where you just kind of hit the ball and then you stop you're not going to see as much force or think about like a pool stick you know when you kind of follow through you're increasing the contact time so the formula again is just impulse well we'll get into it here with changing impulse but uh the next we can also look at it with a you know a high jumper he's going to generate an Impulse with his legs here to push them off off to the ground here's a kick of the football and here's a baseball bat again we kind of went over these uh exactly you know how these worked so now we're going to talk about impulse so to change momentum you have to apply a for force over a given period of time we look at the equation we've been using we can rework that in order to put the time over on the other side so now we're looking at changing forces over or net forces times time will equal a change in impulse okay so uh the more time you add or the higher the force the greater the change in impulse is going to be so impulse is the force multiplied by the time the force is being applied impulse can modify momentum by either a large force over a short period of time or a small force over a large period of time okay um we're going to get into a couple examples here so you can look at amateur golfer uh amateur golfers typically use large forces but they apply them over a short period of time so the goal to of like hitting a ball is to generate as much power as possible one quick motion however because the time of contact of the ball is short it limits how much momentum the ball is going to change from a zero state to a moving state so the object's result is less controlled and has less distance so think about someone who's golf in you know they don't follow through they have very low short contact time but if you have someone who knows how to play they have a lot more contact time with the ball so like a professional gol golf w l large or sometimes even moderate forces and not as high but they sustain them over a longer period of time during their swing so by increasing increasing that duration of force application the professional is able to get the ball farther have you can get more momentum into the ball you know this difference highlights you know how important technique and timing is in sports you know and uh you know like optimizing how much force you can put into something you know to get it to have more momentum by just applying that Force for a longer period of time oh that's that uh really bad uh Will Smith kick I don't know if you guys remember that but he did not kick he did not follow through and it went not great but again you can see these examples here as well like showing how Malong and how the force is being generated throughout the uh body but okay um let's do a quick practice problem um I want to throw a 049 kilogram ball 40.2 m/s uh which is 90 mil hour but must do this Within 2 seconds how much force do I need to apply well first of all let's just write down our equations we need we have change in force times time equals change of mass times velocity or change of you know momentum is really what we're looking at so um Mass really doesn't change so it's this this change here is typically applied to this velocity change of momentum but uh again we can break this down into what we know we know that we have a timing aspect here of 0 26 seconds we'll throw that into the force times time this time segment we know the mass of the object which is going to be you know 0.149 and then we have a change in velocity which is going to be you know four zero you know starting at nothing to 4.23 m per second so now that we know all this we can just quickly do some algebra in order to figure out how much force is needed which is going to be you know 5.99 uh kilog per Meer squ that's going to be how much or it's going to be 22.5 3 sorry this is this is the momentum I was kind of confused for a sec yeah so this is the momentum after we've done the calculations which is just basically multiplying this into this gives us this 5.99 we know the time we divide time to size now we get 22.2 uh 53 so that is how many newtons of force you have to apply to a ball if you only are going to having contact of it with of like applying that ball at. 26 seconds okay uh we have another problem here uh if we decide to apply 15 Newtons to the ball how long should we apply this Force to reach this work the problem but now we're going to look at the time so we just redo the problem here and except this time we don't know the timing aspect of it use some algebra and we come out to be about 04 seconds or 399 seconds after you've done the math so again this impulse momentum equation this momentum equation can be used in multiple different ways but this is now kind of more of the it's getting into more what you'll do in your lab this week um we can use the impulse momentum relationship here to estimate how high can some someone can jump based on the vertical ground reaction forces so something we collected in lab before vertical ground reaction forces remember our first lab so we can actually calculate the average uh Force being applied to it over a period of time in order to understand how much impulse or how what the change of momentum would be and then from there we can calculate their initial velocity and from there we can see how high they can go because we can use some projectile equations as well so this is all kind of coming full circle so again we know this is anything from here and back this is all Ecentric this is all the concentric all this area in here that's all concentric movement that's all you pushing against the ground in order to try to get off the ground so we can calculate you know our average force of this period and we also look at the time how long were you applying that Force into the ground get time in order to see what your change in momentum would be but we'll get into that in a little bit okay so let's kind of use this example here so we have a body weight impulse so this force is the force this is the force due to your own body weight over a period of time so it's basically the impulse of your body experience is just standing there before you even start the jump this is your body weight over time because you're always going to be applying a force on your body over a period of time but so we have to kind of we calculate that as impulse of body weight so let's say your body weight is well we we'll get into it we have a we have an example I'm not going to get into to that right now um then we have a jump impulse so the jump impulse is the extra force that comes from the actual movement when you're applying more than just your body weight this is the force that lifts you off the ground this is you Contracting your legs pushing into the ground in order to get yourself off the ground so in order to get off the ground you have to have a higher jump impulse than your body weight impulse Okay so no nope no no so impulse here is going to be the force over time of uh you pushing and Contracting into the ground the force getting you up into the air it's the force on top of what you're already standing with because you already have a net layer right here you have to push into the ground and try to get trying to get yourself off the ground so now we can kind of look at total impulse so total impulse is going to equal the impulse of your body weight plus the puls of jump impulse so this is the all the impulse that's being applied to your body during this jumping process you're going to be applying a impulse of body weight just by standing there and then also you're going to on top of that be contracting into the ground so this total impulse is going to be the overall force applied to the ground which allows us to estimate things like like a jump height uh how long you'll be in the air what Vertical Velocity you're going to take off with um it's going to be easier once we kind of look at this example question here so we have this jump impulse which equals impulse total the total impulse you're applying to a plate minus the impulse of your body weight okay so jump impulse again we talked about it total plus uh body weight is going to be your jump impul or you're going to be your total so if you rework that equation now you get impulse of just the jump equals the impulse total minus impulse body weight just by doing Simple subtraction impulse of the jump is going to equal to that Force being applied the average force over the time period you're applying that Force um we can even further break that down into mass times change in velocity or you know a change in momentum um so how much are we changing an object's momentum so if the initial uh velocity is going to be zero then our jump impulse is just going to be whatever your final velocity is minus the zero which is just you know your starting speed which again if you're standing there it's going to be nothing so we can kind of ascertain from this equation that velocity final equals impulse of the jump divided Mass okay and then we can use this in order to try to ascertain what your height is because we know your final velocity your final velocity is going to be basically your initial velocity of your jump your projectile and you just throw this into the projectile equation in order to figure out what your height is how high did you go and we've done this before if we know what your initial Vertical Velocity is you can go vertical final being zero in order to find out what you know how high you went are you can use this equation right here too this equation will also work for height but you can use the other projectile equation as well so all right let's use an equ example this is what I wanted to get to okay so we have the impulse of the body weight which is six 620 uh you know8 Newtons per second we have the impulse total Al which is going to be 7 77281 Newtons per and we have a mass of 56.2 so first thing we want to do is calculate your impulse your jump impulse we know jump impulse is going to be the total this number right here how much total you're applying just subtracted by the body weight which is right here so once you do that you get a jump impulse of being 152 Newtons per second okay now we can rework that problem into the other equation we just got which is B final equals impulse jump over Mass which again we calculated earlier and that's going to lead us with a final velocity of 2.7 m/s and then when we want to see what how much we got off the ground we could plug and chug that into this equation here you can also use the other equations there's other equ other than this one in order to get jump height okay this is not the only one but this is one of the equations you can use this is interchangeable um and this is going to give us an answer of 372 Metter okay and we're going to be do using this in lap we're going to be calculating our jump what our average jump impulse is what our average total impulse is in order to figure out what our height would be through a jump and we'll go over that more during our last period okay cool um couple more example questions here we have you know player a u see and player B kind of hitting each other this is that momentum conservation question so a 90 kilogram hockey player traveling a velocity of 6 met per second collides with a headon with 120 kilogram player traveling at 7 m/ second in the opposite direction or negative 7 m/ second what is their combined velocity after they've Tangled together after the Collision so again we they combine together which direction are they moving in at what speed so we have our known we can calculate our in our U momentum from that what is our unknown we don't know our final velocity so we throw into these equations here that we've already done in order to get a final velocity of 1.43 in the direction a player B so the opposite going to the left we'll just say CU he had more momentum which makes sense because he had a higher speed and a higher weight so he's obviously going to have more momentum okay that is the end of this I know this is a complicated concept I'm going to be covering it more in our inperson lecture on Monday uh all right