Acceleration
Why Distance is Area under Velocity-Time Line Understanding why distance is area under velocity-time line
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- Let's add something moving with a constant velocity of 5 m/s
- and we're assuming it's moving to the right
- just to give us a direction, because this is a vector quantity
- so it's moving in that direction, right over there
- And let me plot its velocity against time
- so this is my velocity
- I'm actually going to only plot the magnitude of velocity
- and you can specify that like this: ||v||
- so this is the magnitude of the velocity
- And then, on this axis, I am going to plot time
- So we have a constant velocity of 5 m/s
- so its magnitude is 5 m/s, and it's constant, it's not changing
- as the seconds ticks away, the velocity does not change
- So it's just moving 5 m/s
- Now, my question to you is: how far does this thing travel after 5 seconds?
- So after 5 seconds, so this is 1s...2s...3s...4s...5s...right over here
- So how far did this thing travel after 5 seconds
- Well we can think about it two ways
- 1) We know that velocity is equal to displacement over change in time
- and displacement is just change in position
- so this is change in position over change in time
- or 2) another way to think about it, if you multiply both sides by change in time
- you get velocity times change in time, is equal to displacement
- So what was the displacement over here?
- Well I know what the velocity is, it's 5 m/s
- 5 m/s, that is the velocity (let me color-code this)
- And we know what the change in time here is, it is 5 seconds
- and so you get...the seconds cancel out with seconds...
- you get 5 *5 = 25 meters
- And that's pretty straight forward
- but the slightly more interesting thing is
- is that's exactly the area under this rectangle right over here
- And what I want to show you in this video
- that is in general, if you plot velocity, the magnitude of velocity
- ...so you could say speed versus time...
- or let me just stay with the *magnitude of the velocity* versus *time*
- the area under that curve is going to be the distance traveled (or the displacement)
- because displacement is just the velocity times the change in time
- so if you just take out a rectangle right over there
- So let me draw a slightly different one where velocity is changing
- so let me draw a different situation where you have a constant acceleration
- the acceleration over here is going to be 1 m/s/s, so 1 m/s^2
- and let me draw the same type of graph
- (although it's going to look a little different now)
- So this is my velocity axis
- (I'll give myself a little bit more space)
- So this is my velocity axis
- I'm just going to draw the magnitude of the velocity
- and this right over here is my time axis
- so this is time, let me mark some stuff off here
- so...1...2...3...4...5...6...7...8...9...10
- and...1...2...3...4...5...6...7...8...9...10
- and the magnitude of the velocity is going to be measured in m/s
- and the time is going to be measured in seconds
- So what's going to happen here?
- Assuming that we start with...
- so my initial velocity, or I could say the magnitude of my initial velocity
- so just my initial speed, you could say, this is just a fancy way of saying my initial speed...
- ...is zero
- So my initial speed is 0
- so after 1 second, what's going to happen?
- After 1 second, I'm going 1 m/s faster
- so now I'm going 1 m/s. After 2 seconds what's happened?
- Well now I'm going another 1 m/s faster than that
- After another second, if I go forward in time
- if change in time is 1 second, then I'm going a second faster than that
- And if you remember the idea of *slope* from your Algebra 1 class
- that's exactly what *acceleration* is, in this diagram right over here
- We know that acceleration is equal to...change in velocity over change in time
- over here change in time is along the x-axis
- so this right over here is a change in time
- and this right over here is a change in velocity
- When we plot velocity (or the magnitude of velocity) relative to time
- the slope of that line is the acceleration
- and since we're assuming the acceleration is constant
- we have a constant slope
- so we have just a line here, we don't have a curve
- Now what I want to do is think about a situation
- let's say that we accelerate at 1 m/s^2...and we do it for
- so the change in time is going to be 5 seconds
- and my question to you is: how far have we traveled?
- Which is a slightly more interesting question than what we've been asking so far
- So we start off with an initial velocity of 0
- and then for 5 seconds, we accelerate at 1 m/s^2
- so 1...2...3...4...5...so this is where we are
- so after 5 seconds, we know our velocity
- our velocity is now 5 m/s
- But how far have we traveled?
- So we can think about it a little bit visually
- we could say, look, we could try to draw rectangles over here
- we were at, maybe right over here we had a velocity of 1 m/s
- so if I say 1 m/s times a second, that'll give me a little bit of distance...
- and the next one I have a little bit more of distance...
- calculated the same way. I could keep drawing these rectangles here
- But then you're like...Wait! Those rectangles are missing...
- because I wasn't...for the whole second...I wasn't only going 1 m/s...
- I *kept* accelerating, so actually maybe I should split up the rectangles
- I could split up the rectangles even more
- So maybe I go, every half-second
- so on this half-second, I was going at this velocity
- and I go at that velocity for half a second
- velocity times the time will give me the displacement
- and then I do it for the next half second
- same exact idea here, it'll give me the displacement
- so on and so forth
- I think what you see is your getting...
- is the smaller the rectangles you try to make here
- the closer you're going to get to the area under this curve
- And just like the situation here, this area under the curve is going to be the distance traveled
- and lucky for us, this is just going to be a triangle
- and we know how to figure out the area for a triangle
- so the area of a triangle = (1/2) * base * height
- which hopefully makes sense to you
- because if you just multiply base * height
- you get the area for the entire rectangle, and the triangle is exactly half of that
- So the distance traveled in this situation
- or I should say the displacement
- just because we want to make sure we're focused on vectors
- the displacement here is going to be
- (or I should say the magnitude of the displacement...which is the same thing as the distance)
- is going to be 1/2 times the base...which is 5 seconds
- times the height...which is 5 meters per second
- ...times 5 m/s (let me do that in another color)
- the seconds cancel out with seconds
- and we're left with (1/2) * 5 * 5 meters
- so it's (1/2) * 25, which is equal to 12.5 meters
- and so there's an interesting thing here
- there's a couple of interesting things
- hopefully you realize that if you're plotting velocity versus time
- 1) the area under the curve, given a certain amount of time, tells you how far you have traveled
- 2) the other interesting thing is that the slope of the curve tells you your acceleration
- What's the slope over here (left)?
- Well, it's completely flat, and that's because the velocity isn't changing
- so in this situation, we have a constant acceleration
- the magnitude of that acceleration is exactly zero
- our velocity is not changing
- here (right) we have an acceleration of 1 m/s^2
- and that's why the slope of this line right over here is 1
- the other interesting thing, is that even if you have constant acceleration
- you can still figure out the distance by just taking the area under the curve...like this
- so we were able to figure out...we were able to get 12.5 meters
- The last thing I want to introduce you to...
- (actually let me just do it in the next video)
- and I'll introduce to you the idea of *average velocity*
- now that we feel comfortable with the idea that
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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