Hey, guys. So in your motion problems, you're going to run across this phrase, "speeding up" or "going faster," and you might take that to mean that your acceleration is positive, but be careful because this does not always mean this. Students get tripped up all the time. So in this video, I'm going to show you how the acceleration sign affects velocity and speed. I'm going to show you a really simple way to think about this. Let's check it out. So guys, again, positive acceleration does not mean speeding up or going faster. Positive acceleration literally just means that your velocity is becoming more positive. And on the other hand, negative acceleration just means that your velocity is becoming more negative. And we can just see this from the equation. We know that acceleration is ∇v∇t. So positive acceleration means that the change in velocity is positive, aka your velocity is becoming more positive. And then negative acceleration means that your velocity is becoming more negative. That just comes from the equation. Let me just show you a bunch of examples to see how this works and where the confusion comes from so you don't make the same mistakes yourself.
So we have all these problems here. We're going to indicate whether the acceleration is positive or negative and if the speed is increasing or decreasing. We're going to assume ∇t = 4. So let's get to it. We've got vv0=10, vfinal=30. So I'm just going to draw a quick little sketch. Here's my v0 which is 10. Then at some later time from here to here, I know that my final velocity is going to be a bigger vector, and this is going to be 30. So to calculate the acceleration, I need ∇v∇t. So my change in velocity is 30 minus 10, that's my initial divided by ∇t which is 4. So this is 20 over 4, and I get 5 meters per second squared. So notice how this is a positive number, that's my acceleration, so that's positive. What about the speed? Well, whenever you think about the speed, just think of the number of the velocity, the magnitude of velocity, not the actual sign. So I'm going from 10, that's this number, to 30, which is a bigger number. Again, forget about the sign, which means that my speed is increasing.
Let's take a look at part b. Now I have a velocity that's negative and a final velocity that's negative. So I'm actually going to start going to the left. So this is my initial velocity here. My v0 is negative 10. And then my final velocity over here is going to be negative 30. And I know that this takes some time, which is ∇t. So to calculate my acceleration, I need ∇v∇t. My ∇v is going to be negative 30, that's my final minus my initial negative 10. Be careful with the signs. And then ∇t is 4. So this is actually negative twenty over 4. So this is actually negative 5 meters per second squared. So my acceleration's negative. So this is my acceleration over here. What about the speed? Well, again, just think about the number, not the sign. Just think about the magnitude of your velocity. I'm going from negative 10, so the number is 10 to negative 30. The number is 30. The number gets bigger. Forget about the sign. So that means my speed is increasing. So notice how in one situation, my acceleration was positive, my speed was increasing. And in the other situation, my acceleration was negative, but I was still speeding up, and my speed was still increasing. So speeding up just means that you're going faster. And that means that the magnitude of your velocity, just think about the number, is increasing. And so the easiest way to think about this is if your acceleration and your initial velocity are of the same sign, that's when your speed increases. For example, here, my acceleration was positive. My initial velocity was positive, so my speed increased. Over here, my acceleration was negative. My initial velocity was negative, and so my speed increased as well.
Let's take a look at more problems. So here we've gotten a velocity of 30 and a final velocity of 10. So here, now I've got my initial velocity is 30, and then my final velocity is going to be 10 over here. So I'm going to draw up my little interval from here to here. Now, I just calculate the acceleration. A=∇v∇t. My ∇v now, my final velocity is 10, my initial velocity is 30, and my time is 4. So this is going to be negative 20, and this is going to be 4. So I get an acceleration of negative 5 meters per second squared. So my acceleration is negative. What about the speed? Again, think about the number. First, I'm going from 30, and now I'm going to 10. The number is getting smaller this time. So my speed is actually decreasing. Let's go to the last one, negative 30 to negative 10. So here what happens is, first I'm going, this is my v0, it's negative 30. Draw it again. Negative 30 over here. And then finally, it's going to be negative 10. And then my interval is over here. I know that's 4 seconds. So what's my acceleration? So acceleration, ∇v∇t. So my ∇v, what's my final velocity? It's negative 10. My initial velocity, negative 30. Again, keep track of all the signs. Over 4. So this becomes positive, and positive. This becomes 20 over 4, and that's just 5 meters per second squared. And it's positive. So here we got a positive acceleration. What about the speed? The speed is going from 30 to 10, the number, just forget about the sign, the number is going from 30 to 10, which means that my speed is actually decreasing here. So again, now we have a situation where we have negative acceleration and decreasing speed, but positive acceleration could also mean decreasing speed. So slowing down just means that obviously, you're going slower, which means that the magnitude of your velocity is decreasing, and this happens whenever your acceleration and your initial velocity have the opposite signs. So for example, here, my acceleration was negative, positive initial velocity, so I slowed down. Here, my acceleration was positive, initial velocity was negative, and so that means I am also slowing down. Alright, guys. Hopefully, this illustrates where that confusion comes from and so hopefully, these rules here give you a better understanding of what the relationship between signs of acceleration and velocity are.
Alright. I've got this one last problem here. Let's just get to it. We've got the driver of a truck that is moving to the left at 30 meters per second, slows down by taking their foot off the pedal. And basically, we're going to calculate the magnitude and direction of the acceleration, and it's assumed constant. So we're actually just going to use our normal steps for solving motion problems with acceleration. So let's just get to it. Let's just draw a quick little sketch here. So I've got this truck that's moving to the left. So here, I'm at this truck that's moving to the left. We know that this initial velocity here is going to be 30 meters per second, but if it's moving to the left, it's actually going to be negative. And so, we're told that the truck comes to a stop after traveling 150 meters. So basically, after some distance over here, which we know ∇x is going to be negative 150. Now, at this point here, the final velocity is going to be 0. And again, this is negative because we're moving to the left. So we need 3 out of 5 variables. Let's just go ahead and list them out. I need ∇x, I need vnaught, v, a and t. I know ∇x, is negative 150. I know my initial velocity is negative 30, and my final velocity is 0, and then my a is what I'm looking for. This is my target variable and then, yeah. So I've got my diagram, and I've got my 5 variables. I've got the known variables and what my target variables are. So now, I just have to pick the UAM equation that does not have my ignored one. So these are my 3 variables that I know. I'm looking for the acceleration, and the time is going to be my ignored variable. So if you look through our list of equations here, the one that doesn't include time is actually going to be the second one over here. So we're going to use equation number 2. Whoops. So vfinal2=vinitial2+2a∇x. And so we're looking for this acceleration over here. We know that my final velocity is going to be 0, 02. My initial velocity is going to be negative 30, but it's squared, and then plus 2 times a times ∇x. So I get 0 equals this, being 900 + 2 times a times my ∇x, which is negative 150. So this ends up being, when I move this to the other side, I get negative 900 equals, and then when I multiply 2 and the negative 150, this becomes negative 300 times a. And so, all I have to do now is divide, and basically, my a is going to be 900 over 300, which is just 3 meters per second squared. What about the sign? Well, now my sign is positive. It's a positive 3 meters per second squared, but notice how everything in my diagram was negative. So even though I was slowing down by taking my foot off the pedal, my acceleration was actually positive, and that's because everything in my diagram was pointing to the left. Alright, guys. That's it for this one. Hopefully, this sort of clears up some of the confusion. Let me know if you have any questions.