Impulse and momentum dodgeball example

this person right here is about to play dodgeball they're just unfortunately not going to dodge the ball it's going to fly in it's going to bounce off their head this may or may not have happened to you I think it's probably happened to me I spent a long time since I played dodgeball and although unfortunate for this person it's a wonderful opportunity scientifically speaking to talk about the impulse momentum force time relationship so let's do that let's put some numbers on here so we're going to need to know the mass of the ball let's say this is a point two kilogram ball and we're going to need to know some other numbers let's say the ball comes in at a speed of about 10 meters per second so it says coming in at 10 meters per second let's say it leaves at a speed of 5 meters per second so it's probably gonna recoil with a little less speed than it came in with so comes in with 10 leaves with 5 and let's say the time the time right here the time period that it's actually in contact with the person's face let's say the time when the ball is getting kind of compressed and then recoils and expands again let's say the time that it's actually in contact is about 0.02 seconds or about 20 milliseconds so knowing this information we can ask all kinds of questions one of them is what was the impulse on the ball from the person now the definition of impulse we use the letter J for impulse that always seemed a little weird to me there's no J in impulse I end up calling it Jim pulse just so I could remember that it's impulse and there's a J for it so the Jim pulse or the impulse is defined to be the force acting on the object multiplied by the time duration during which that force is acting in other words the impulse from a force is equal to that force multiplied by how long that force was acting on the object so if we knew the force on this ball we could use this formula to get the impulse but we don't I don't know the force that this person's face is exerting on the ball so I can't use this formula to solve for the impulse but there's an alternate formula for impulse if you're talking about the net impulse in other words the impulse from all forces on an object like this ball that should just equal the change in momentum of that object like the change in momentum of this ball so if we can figure out the change in momentum of this ball we could figure out the net impulse on this ball and since it's the net impulse and this formula pier is also true this is equivalent to just saying that it's the net force multiplied by the time duration during which that net force is acting this is hard for people to remember sometimes my students like to remember it as jape fad so if you look at this it looks like J a P this kind of looks like an e f80 so if you need a way a mnemonic device to remember this jape fad is a way to remember how impulse change in momentum force and time are all related so let's do it we can't use force because we don't know yet but I can figure out the change in momentum because I know the velocities so we know that the change in momentum is going to be P final the final momentum minus the initial momentum what's my final momentum my final momentum is M times V so it's going to be mass times V final minus mass times V initial and my mass is 0.2 so I've got a mass of 0.2 kilograms my final velocity is 5 because the ball recoiled to the right with positive 5 positive 5 because it's moving to the right and I'm going to assume rightward is positive and then minus the mass is 0.2 again so 0.2 kilograms my initial velocity is not 10 okay this is 10 meters per second to the left and momentum is a vector it has direction so you have to be careful with negative signs here this is the most common mistake people just plug in positive 10 they get the wrong answer but this ball changed directions so the two velocities here have to have two different signs so this has to be a negative 10 meters per second if I'm assuming rightward is positive this leftward velocity this leftward initial velocity has to be negative 10 and if you didn't plug that in you'd get a different answer so you got to be careful so what do I get here if I multiply this all out I'm going to get zero now sorry I'm going to get one kilogram meter per second minus a negative two kilogram meter per second and that's going to give me positive three kilogram meters per second is the impulse and that should make sense the impulse was positive the direction of the impulse which is a vector is the same direction as the direction of the force so which way did our face exert a force on the ball our face exerted a force on the ball to the right that's why this impulse on the ball is to the right the impulse on this person's face is to the left that the impulse on the ball is to the right because the ball was initially going left and it had a force on it to the right that made it recoil and bounced back to the right that's why this impulse has a positive direction to it now if you've been paying attention you might be like wait a minute hold on what we really did was we found the change in momentum of the ball and when we do that what we're finding is the net impulse on the ball in other words the impulse from all forces on the ball but what this question was asking for was the impulse from a single force the impulse from just the person's face now aren't there other forces on this ball isn't there a force of gravity and if there is doesn't that mean what we really found here wasn't the impulse from just our face but the impulse from the person's face and the force of gravity during this time period and the answer is no not really for a few reasons the most important reason being that what I gave you up here was the initial horizontal velocity so this 10 meters per second was in the X direction and this five meters per second I'm assuming is also in the X direction as if I'm taking the initial velocity in the X and the final velocity in the X and I take the difference in momentum what I really found was the change of momentum in the X direction when I do that I'm finding the net impulse in the X direction and there was only one X directed force during this time and that was our face on the ball pushing it to the right there was a force of gravity that force of gravity was downward but what that force of gravity does it doesn't add or subtract any impulse in the X direction it tries to add impulse in the downward direction in the Y direction so tries to add vertical component of velocity downward and so we're not even considering that over here we're just going to consider that we're looking at the horizontal components of velocity how much philosophy does it add vertically gravity typically not much during the situation because the time period during which this collision acted is very small and the weight of this ball compared to the force that our face is acting on the ball with the weight is typically much smaller than this collision force so that's why in these collision problems we typically ignore the force of gravity so I have to worry about that here that's not actually posing much of a problem we did find the net impulse in the x-direction and since our face was the only X directed force this had to be the impulse that our face exerted on the ball now let's solve one more problem let's say we wanted to know what was the average force on this person's face from the ball well we know the net impulse on the ball that means we can figure out the net force on the ball because I can use this relationship now so since I know that the net impulse on the ball in the X direction should just equal the net force on the ball in the X Direction multiplied by the time interval during which the force was applied I can say that the net impulse on the ball was three kilogram meters per second and that should equal the net force on the ball in the X direction which was supplied by it unfortunately this person's face multiplied by the time interval which is 0.02 20 milliseconds so now I can solve the force the net force on the ball during this time interval and the X Direction was 3 divided by 0.02 so if I take this if I take 3 kilogram meters per second and I divide by 0.02 seconds I'm going to get 150 Newton's was the net force on the ball we got a positive number and that makes sense because this person's face exerted a positive force on this ball because the force was exerted to the right so these were positive and the impulse from the face on the ball should go in the same direction as the force from the face on the ball so this is the force on the ball by the person's face but notice this question is asking what was the average force on the person's face from the ball not on the ball by the face you might think god no I got to start all over we solved for the wrong question but we're in luck Newton's third law says that the force on the face from the ball should be equal and opposite so this force on the face from the ball has got to be equal and opposite to the force on the ball from the face so that's what we found here the force on the ball from the face that means the force on the face from the ball is going to have the same size it's going to be 150 Newtons it's just going to be directed in the leftward direction that means it's going to be a negative force so technically you could say this would be negative 150 Newton's on the face from the ball so to recap the impulse from an individual force is defined to be that force multiplied by the time interval during which that force is applied and if you're talking about the net force in a given Direction multiplied by the time interval you'd be finding the net impulse in that direction and this also happens to equal the change in momentum in that direction so in other words if there is a net impulse in a given direction there's got to be a change in momentum in that direction by the same amount and one convenient way to remember how are these related is you could use the mnemonic device Jade fat I have no idea what jape fad means but it helped me remember that the net impulse equals the change in momentum and that also equals the net force multiplied by the time interval during which that force was applied and finally remember that during these collisions there's always an equal and opposite force exerted on the two objects participating in the collision