Impulse of a Baseball Bat

by Patrick Ford
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Hey everybody. So in this problem we have a baseball bat that's hitting a bat and we have this graph here that shows the force over time. So it's a massive amount of force. But it happens with a very very small amount of time. The scale is in milliseconds here. Let's jump into the first part here. We want to calculate the impulse that is delivered to the baseball. So in other words we want to calculate is j remember j whenever you're given a force versus time graph is it's going to be the area that is under the curve. So in other words, you just take this graph here, you split this up and do a bunch of triangles and squares and stuff like that. And you just calculate the area that's under the curve. So this shape is kind of like a triangle but it's kind of weird because this side here is a little bit different than the right side. But what we can do is we can break it up into a smaller to smaller right triangles. And I'm gonna call this one a one and a two. So to calculate the impulse, we're gonna calculate the area and that's really just the area of a one plus a two. All right, so let's go ahead and write out these formulas here. Um Well a one is gonna be this has this is a base, I'm gonna call this base one and this has a height of h the other triangle has a base. I'm gonna call base to and it also has the same height. So it's the H. This is the same age for both of the triangles. So that means that the area formula for area one is going to be one half of base one times height plus one half of base two times heights. And what we can do here is we can just do a little math or geometry trick, which is we're gonna combine these two bases together. This is gonna be one half, base one plus base two times height. So really all you need to calculate for any triangle doesn't necessarily have to be a right triangle is you need to base the whole entire thing times the heights and then you multiply it by one half. Alright, that's just a cool little shortcut. But anyways so we're actually going ahead and plug this in. So we've got the base one which is gonna be what are we gonna plug in four. Well remember that this scale here is in milliseconds and we have to convert it to the right units. Four milliseconds is really just 0.4. So be careful here because this four here is going to be 0. seconds. Likewise, what happens here is we have to do the base too. This is going to be 8:00 which is going to be 0.008. So we add that in 0.008. And then you have to multiply by the height of the whole triangle which is 115 100 when you work this out, what you're gonna get is an impulse of nine newton seconds. Alright. So even though the force is massive it's 1500 it acts over a very small amount of time. And so you end up with something you know like nine which is a super huge number. Alright, so that's the answer. That's the impulse. Let's move on to the second part here. Now we want to calculate the final speed of the baseball right after the impulse, right? So after the bat hits the ball and it goes off, we want to calculate what is the speed after that impulse is delivered? So how do we do that? We'll remember that the impulse can only always be related to the change in the momentum. Another formula that we have here is that J. Is equal to force times time, but it's also equal to basically the change in the momentum. So we have M. V final minus M. V. Initial. What we really want to calculate here is the final. So if you look through what happens is that the second term, the mv initial, this is the initial momentum is going to be zero. What happens here is that there is no velocity, there's no initial velocity because the baseball is initially at rest, right? So there's no initial speed here. So the whole entire term goes away. So what's the J. Well we just calculated it, it's the nine newton seconds. So this is gonna be nine is equal to and then we've got the mass. The mass is 200 g. So this is gonna be 0.2 kg. This is gonna be zero point to the final. So if you work this out, what you're gonna get here is nine divided by 0.2. And that's the final speed of 45 m per second. And that's pretty reasonable because this is about, you know, let's say 100 miles an hour and that's pretty reasonable for a baseball, right? You can get, you know, you hit the baseball and it goes off like 100 miles an hour. That's someone reasonable. Alright, So that's the answer. Let me know if you have any questions.