Velocity time graphs (& acceleration)

- A cheetah is running somewhere and you would love to see it. But since you like physics so much, instead of showing you the clip, I give you it's a velocity-time graph. Whew, So let's see if we can analyze this graph and visualize what our cheetah is doing. So, what's a velocity-time graph? It's a graph that tells us the velocity of any object, like our cheetah, at every moments in time. I like to think of it as watching a speedometer of a car. It is just like that. The speedometer tells us what the speed of our vehicle is at every moments in time. The only difference is, here, it not only gives us the speed, not only tells us how fast it's going, but it also tells us in what direction that object is moving. That's what velocity is. It's speed plus direction. How does this graph give us a direction? Well, if the graph is positive, like right now, like you can see over here about the time axis, then we can say the object is moving in one particular direction, let's say to the right, or maybe upwards as an example. And if the graph were to come below the t-axis, then we would say the velocity is negative. Meaning it's in the opposite direction. It could mean to the left or it could mean downwards, anything. And so the graph gives us both the speed and the direction. That's why it's called velocity-time graph. But let's not worry too much about the direction, we'll only concentrate on the speed part. So what is the graph saying? Well, the graph is saying that at time, t equal to zero, our cheetah has zero speed. But then as the time ticks, look at what's happening. It's speed is increasing. Can you see that, one second it's speed is increased. It's speed increases. So it's speed is continuously increasing and at the end of five seconds, it has reached 20 meters per second. So if we had a speedometer attached to our cheetah, then it would look somewhat like this. At time, t equal to zero when we just start our timer it has zero speed, but then it quickly accelerates. Its speed increases and at the end of five seconds, it reaches a spirit of 20 meters per second. And we don't know what happens after that because the graph ends over there. And now I'm guess we can pretty much visualize what our cheetah is doing. So, let's say here is our one of a kind pink cheetah. We know that at time, t equal to zero it's speed is zero. So if we attach this speedometer to it, then as time ticks, it gets faster and faster and faster, and at the end of five seconds, it hits 20 meters per second. Now, one important thing to think about is that just like when you're looking at the speedometer of a vehicle, it only tells us how fast it's going, but it doesn't tell us where that object is, or where that vehicle is, right? Similarly, the velocity-time graph, also tells us only the speed, how fast it's going, and it tells us in what direction is going. But it doesn't tell us anything about its location, its position. So I have no idea where this cheetah was to begin with. It could have been at zero, but it could have been a hundred meters or maybe it would have been that minus a hundred meters. There is no information about its position, but only about its speed and direction. Okay, now that we know our cheetah is getting faster and faster, it means it's accelerating, right? Whenever objects change their speed, whether they increase it or decrease it, we say that object is accelerating. So, can we calculate this acceleration by looking at this graph? Well, let's see. How do we calculate acceleration in general. We define acceleration, a, as the change in velocity divided by time. Change in velocity is always calculated as the final velocity, v minus initial velocity, u. So this is the change in velocity divided by time. So let's see if we can look at our graph and calculate this. In fact, you know what? See if you can try this on yourself first. Go ahead, pause the video and see if you can do this. If you can calculate the acceleration by looking at the graph. All right, let's see. We know at t equal to zero, it's velocity is zero. Let's call that out as our initial u. And we know at five seconds, its velocity is 20. Let's call that as v, that's our final. And so acceleration becomes final velocity minus, it's meters per second, minus initial velocity, which is zero. Do you have the time taken for that change? And the time taken was five seconds. And that number equals 20 divide by five. So it's five seconds. 20 divided by five, that's going to be four meters per second per second, Which we call four meters per second squared. That is the acceleration of our cheetah. What does it mean? It means our cheetah is gaining a speed of four meters per second every second. Every second it's increasing its speed by four meters per second. That's the acceleration of our cheetah. Okay, now let's say there's a deer that is chasing this cheetah. Whew! Things are getting very interesting. And it's velocity-time graph is over here. We see that even this deer is accelerating. Now, can you figure out what is the acceleration of this deer? Is it more than the cheetah or less than that? Again, pause the video and see if you can do this. Okay, let's see. Again at time zero, our deer's also be zero. We can call that as initial. And then at the end of two seconds, it reaches 20. So we'll call that as final. So the acceleration for our deer is going to be v minus u by the same formula. 20 minus zero divided by two seconds, because it reaches 20 in only two seconds. And so that would equal 20 divided by two. That is 10 meters per second per second, or 10 meters over second squared. which means the deer has a way higher acceleration compared to the cheetah. That makes sense, right? The cheetah took five seconds to increase its speed to 20 and our deer only took two seconds to increase its speed. So it has a higher acceleration. And in this crazy world, because our deer is chasing the cheetah, can we say that the deer will quickly catch up to it? Need not be. Because remember, we don't know anything about their initial locations, initial positions. It could be totally possible that the cheetah is miles ahead of the deer to begin with. So we can't comment on how long it would take to catch up at all. Another most important thing we can see from the graph is that the deer which has a higher acceleration has a much steeper graph compared to the cheetah. I mean, if you were to think of these as mountains which you are climbing, then notice, climbing this mountain is much steeper. It's much harder than climbing this one, right? And so this means that in a velocity-time graph, if the graph is steeper, more acceleration. If the graph is less steep, less acceleration. And that's why we like this graph because just at a glance, by looking at the steepness, we can figure it out, which has more, and which has less acceleration. And this also helps us understand one more thing. Again, if you were to say climb this mountain, then regardless of where you are, whether you are over here or here or here, the steepness remains exactly the same, right? I mean it's not any harder to climb the mountain here compared to over here. Since the steepness remains the same and the steepness represents acceleration, we can also say that the acceleration of that cheetah is a constant. Meaning, the cheetah is gaining velocity. Its velocity is changing, but it's acceleration does not change with time. It means every second it will gain four meters per second of velocity, constant acceleration. Similarly, even for the deer, it's a constant acceleration. But on the other hand, if the graph was not a straight line, but instead say it looked like this, then it's not a constant acceleration. I mean notice in the first two seconds, it gains a lot of velocity, very high acceleration. The next two seconds, it gains a little bit. The acceleration has decreased. And then after four seconds notice, it's not gaining anything. The velocity remains 20. Acceleration becomes zero. Even if you look from the steepness point of view, can you see that initially it's very steep. So high acceleration, less steep, low acceleration. And then the acceleration becomes zero. The graph is flat. Acceleration is zero. And so if you had a graph that looks somewhat like this, the acceleration is not a constant, it's initially very high and then the acceleration decreases. But for a straight line, it means the acceleration remains a constant. And to understand the difference even more let's bring back our speedometer. So, if you could look at the speedometer for this kind of motion, we would see initially the stick moves very fast because the pickup is very high and then it slows down. Okay, here it is. Notice, initially it was very fast, later it slows down, slows down. And then it's almost a constant. But if we again look at the speedometer for this kind of motion, well notice the pickup is uniform. Can you see that? It's uniformly picking up. And it's for that reason this is also called uniformly accelerated motion. That means its velocity is changing, but it's acceleration is a constant. And so what did we learn in this video? We learned how to analyze the velocity-time graphs. We saw that the steeper the graph more is the acceleration. And we also saw it does not tell us about the position of an object. It doesn't tell us where the object is to begin with, but it only tells us about how fast things are moving.