chapter 7 impulse and momentum so up to this point we've taken a look at forces we've considered that they're constant for all of time or at least all of the interest of the problem right maybe it's when you're driving a car around a turn and we're going to assume the force is constant for that entire turn but in reality a lot of forces are only applied for a short amount of time such as a baseball striking a bat or the bat striking the ball depending on your perspective right the the bat is only in contact with the ball for a short amount of time and in this case the force the bad applies is not constant right i i don't have a baseball bat but you can imagine my ruler's a bat and the tennis ball is a baseball and when they first make contact they're barely touching each other right for a brief nanosecond we're talking tiny scale here and so there's basically no force from the bat to the ball but then as the bat continues to swing through it applies more and more force to the ball until the ball goes from having a velocity going one way to actually taking the velocity and going the other way but if we looked at the force over time like if our bat had a force sensor on it we'd see that the force goes from zero to a very small value shoots up to a maximum value and then decreases again as the ball starts to move away from the bat this is something if you look at a slow-mo captures of it you'll see the baseball actually flattens against the bat it's quite dramatic this happens even with golf balls right which seems so hard if you look at the slow motion video the ball flattens out when when the bat or the club is exerting the largest force on it and then it springs off of the club so the force there is not constant that's a little bit harder to work with rather than trying to work with this weird curvy function right which we could come up with some equation for that but it would be a bit involved and every force would be slightly different we can instead focus on this blue line which is the average force applied during this time interval from the initial time to the final time so notice it's not as large as the largest amount of force but it's more than the smallest amount and this would be if instead of the force having this variable nature if it was constant for that time interval this would come out to the same effect and so this allows us to help describe this we define something new called impulse so impulse or the impulse of a force is the product of the average force and the time interval during which that force acts and we call the impulse j and then we have average force as the bar over it to show that that is average it's not instantaneous times delta t the time interval note that both the force and the impulse have arrows on top so these are vector quantities and so the impulse has the same direction as the average force so with the bat striking the ball since the force on the ball is going that way the impulse is going that way too the units of impulse are just that of force times time or newtons times seconds so with this in mind we could go back to our baseball analogy right and say that okay we're going to describe the impulse that the bat makes on this ball so the bat is applying an average force over a time interval and notice that average force that the bat applies is enough to change the velocity of the ball from going one way to very much going the other way with a different speed as well so we want to connect our ideas of impulse to speed another thing to think about here is if you have say the same speed of pitch the same batter the same bat you're going to get a very different final velocity if it's instead a softball compared to a baseball why would it be different well softball is a lot bigger and it has more mass so that's really something we need to bring into the picture because we want to know how the velocity will change right this is something in the major leagues of baseball they care very much about how fast they can hit the ball because that's going to determine how far they're able to hit if they're hitting a home run or not so we're going to do a little bit of a derivation here but it's pretty short so if we go back to newton's second law our favorite equation f equals m a as you see it just keeps popping up we're going to work with the average force is equal to the mass times the average acceleration and our definition of the average acceleration is the change in velocity we final minus the initial over time all of these are vector quantities we are paying attention to the directions so we can plug that in the definition of acceleration into that and then we have mass times velocity final minus mass times velocity initial divided by delta t is equal to the net the net average force now if we multiply the delta t across notice on the left side here this is looking a lot like our impulse right it's the average force times the time interval that's our impulse on the right we have the final velocity and the initial velocity which is good because that's part of what we're interested in figuring out more about we also have the mass which as i mentioned mass will make a difference in how fast it's going the ball is able to travel after being hit by the bat so we're going to take a moment before we get to a final conclusion to define what is this quantity of mass times velocity we're going to give it its own name because it does pop up a lot this is going to be called linear momentum we're going to also see angular momentum later on so the linear version is just traveling in straight line paths not in a circle or spinning so the linear momentum of an object is the product of the object's mass times its velocity this is a really handy equation to note so we use a lowercase p for linear momentum this is different than uh rho or a capital p for pressure we have a lot of overlap with all the different symbols so note this is a lowercase p the way that i like to draw it is i like to include a little stem over the top and then a flat bar on the bottom just to make it extra clear that it's a lowercase p so that for me is different than my uppercase p which would look something more like this but we're not worried about uppercase here there we go okay so another thing to note is this is a vector quantity right we have vector symbols over the momentum as well as the velocity and because we have vectors over both the linear momentum has the same direction as the velocity so if the velocity is going one way the linear momentum is going that way too what are the units of linear momentum well we don't have any fancy units for them it's just going to be a combination of what we have here mass times velocity so mass is in use of kilograms times our units of velocity meters per second right so now that we've defined this thing called momentum which we will see later on is a really powerful concept and useful for approaching a lot of different scenarios we can go back to that equation that had impulse and see how this all ties together so the impulse momentum theorem says that if we have a net force acting on an object over some time interval the impulse of this force the force times the time interval is equal to the change in the momentum of the object right and find m times velocity final minus m times velocity initial instead of writing out that out either every time we can write that as momentum final minus momentum initial and i didn't have an easy way to make vector symbols but i bolted them to represent the vectors i'll add those in just to make sure to have those so this says whatever your impulse is that describes how the momentum can change so if the baseball bat doesn't apply a net force on the baseball that momentum of the ball is not going to change it's going to keep traveling in that original velocity direction we have to be able to apply a force in order to change it right either to slow it down slightly at least or to even change the velocity of the ball and the momentum of the ball so that it goes in the opposite direction this incidentally also shows another reason why they encourage players to swing through one if you try and stop yourself you're going to slow down your motion and cut yourself short but two if you're swinging through you're maximizing the time that the bat is in contact with that ball so the more time you have in contact with the ball the more impulse you're giving to that ball the more you can change its velocity so that it is able to travel faster which is usually the goal right so this introduces the idea we're going to see a lot more with momentum to come so that is coming but first let's take a look at an example it's an example of a rainstorm rain comes down with a velocity of negative 15 meters per second and hits the roof of a car the mass of rain per second that strikes the roof of the car is .060 kilograms per second assuming that rain comes to rest upon striking the car find the average force exerted by rain on the roof right so we have these raindrops that are coming down and striking the top of our car and it mentions that the rain comes to rest so it just turns into a little puddle here all right we could say v is equal to zero that's going to be our v final specifically go right and there's some other speed beforehand that's going to be our v initial ah it tells us that it comes down with a velocity of negative 15 meters per second v0 is negative 15 meters per second so that's a great thing to write down is what we know they also give us this .060 kilograms per second 0.060 kilograms per second so what do we think this corresponds to it's kind of a mysterious quantity but the units can give us a clue it says kilograms per second so what has units of kilograms well for units of kilograms we have mass what about units of seconds that's our time so this is our mass per unit time is equal to 0.060 seconds kilograms per second that's what they try and say in this statement up here the mass of rain per second so mass per second is going to be mass per unit time all right so this is looking pretty good so far what are we trying to solve for though what do we want to know just to find the average force exerted by rain on the roof so we're looking for average force of rain on the roof all right so as we're thinking about this we can think about what tools do we have we just got one that impulse momentum theorem right that the impulse is equal to the change in momentum so our impulse well i'll just rewrite it in poles equals change in momentum delta p awesome so our impulse is the force times the time interval right and it's really the average force there a vector that's equal to m v final minus m v initial now we don't really know our mass here this could be worrisome except for that we know the mass over the time and we have time on the left side so notice if we want to solve for the force we could just divide both sides by this time interval and then we can distribute that division by a time so it's just applying to the mass and this starts to become much much better looking so our force average force is equal to our mass over our delta t times our v final uh minus mass times the initial the mass is divided by delta t and we know the mass over time and we know the final velocity and the initial velocity so at this point we are in really good shape right so let's check ahead to what they have for the final solutions all right so sure enough they solve for that the final velocity is zero so we just get the the force is negative the mass over the time interval times the initial velocity so we have a negative times this point zero six zero kilograms per second times negative 15 meters per second the velocity is a vector so we have to include that negative sign that says the velocity is going down initially and this comes out to positive 0.90 newtons now why is the force positive i'm glad you asked it's positive because over here this is the force on the raindrop due to the roof right so what the raindrop feels from the roof is an upward force it's gonna it's like running into the wall that's why it's in the positive direction now this describes neglecting the weight of the raindrops that the net force on the raindrop is simply the force due to the roof it's not really relevant to this problem because it already said to just focus on the force from the roof but you could calculate the weight of the raindrop as another four second on the raindrop it just ends up being tiny compared to this right because a single raindrop is like point zero three grams so if you convert to kilograms and multiply by 9.8 to get the weight you get something even smaller it's like .0003 newtons so it's much much smaller than this which is one reason why you could say yeah we could neglect the weight of it even though we don't have to for this problem because it already specified what we're solving for all right so with this in mind let's take a look at a conceptual example so conceptual example here says hailstones versus raindrops instead of rain suppose hail has fallen unlike rain hill usually bounces off the roof of the car if hail fell instead of rain would the force be smaller than equal to or greater than that calculated in the previous example so i want you to think about this choose one of the answers before you move on including the reason why and then move on so go ahead and pause the video now right did you do that okay well let's think about this with the raindrop the final velocity is zero with the hailstone it says that it bounces in the other direction so we have a velocity that now is not zero it's a non-zero value it might be closer magnitude to the speed of the initial velocity but it's in the opposite direction so this is a much greater change in velocity to go from going one way to go in the other way so what we're going to see is that this one has a much greater force in the case of hail than rain and you could show that mathematically you can make up some numbers here that's often a good strategy say that the initial velocity is two and the final negative two the final velocity is positive one and so then you would find if you do final minus initial that comes out to three meters per second times whatever the mass is versus with the coming to rest you would just find a change in velocity of 2 m that's something i encourage you to give it a shot the other way is you can think about it and if you think about going out when there's hail it's a little bit scary because hail hurts because why well hail is hard when it's that hard because it's bouncing your head is exerting a larger force on it as is it exerting a larger force on your head so that change in momentum of it coming down and then bouncing up off your head or whatever else is going to be a more painful force something that you definitely feel right so we'll wrap up there but this introduces the impulse momentum theorem