If this turtle didn't goįorward, down, and up, what did this turtle do? We'll start at t equals zero. So let's just read this graph and figure out what this So somewhere at like two and four seconds this turtle was at two meters.
What that means is if you find the turtle at some point over here at x equals two, then the graph should represent that the turtle is at x equals two by showing the value is two. In this case I wrote is as x, so this is gonna be And I'm doing that because look over here we're graphing. So I'm gonna label this x and it's gonna be measured in meters. Graph actually says let me lay down a horizontal access here. So maybe the turtle went forwardĪnd then down and then up, but that's not right. The first mistake a lot of people make is they think that well maybe the shape of this graph is the same as the shape the And this graph represents the motion of this particular turtle. And I don't want a sternly worded letter. And instead of just saying object, let's make it specific. To write an equation or say a bunch of words. Basically specify theĮntire motion of the object, and you didn't even have Why do so many people love these things? Because you could compactĪ ton of information about the motion of an object into the small little space right here. Though Quantum mechanics is another game altogether, but I digress. You can't have two states of motion in the same instant well, at least not in Newtonian mechanics. Any amount of change, no matter how large or small, takes over some finite amount of time, however small the time interval may be. This can't be possible in the real world. Note that from time t = 0 to an instant before t = 2, the turtle was stationary but as soon as t = 2 seconds, the same stationary turtle now also seems to have a huge velocity along the negative x axis. What the graph is showing is not physically possible and the particle fails to have an instantaneous velocity at the corner where the displacement function is not a smooth curve but a sharp turn. There will always be a smooth curve that would indicate that you safely turned the car left. What does this all mean in the real world? It means you can't be driving your car north at one instant and magically switch direction toward the west in the next instant. If you draw the tangents on both sides of the point, you get two lines headed in different directions and therefore failing to be a tangent by definition. The function is continuous but the limits of the slopes at the points in question doesn't exist because the left-hand limit and the right-hand limit are not equal. " At both 2 seconds and 4 seconds, the limits of the graph exist, but not the limits of the slopes at those points (am I stating this part correctly?)." So it's like the turtle started his race ahead of the starting line, turned around went back behind the starting line and traveled a magnitude of 8m in the process. The distance traveled is only the x direction (the y axis on the graph- admittedly confusing, but it is the independent variable) so from the 2sec to the 4 sec points on the graph the x variable changes 8 points total (from positive 3 to negative five). The underlying principle is to use the simplest model (a point mass) unless further detail is required (For example: if the turtle were racing a hare and we needed to know who crossed the 5m mark first, we'd use the head as it's the most forward part of the turtle's body).Ī3.
measure from when his nose crosses the 3m mark or his tail, but not both). From center of mass (note the white line through the turtles shell) is typically used, but as long as consistency is maintained any singular part of the turtle would work (i.e. for example it could be 3m from a starting line on a track.Ī2.
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