2. 1D Motion / Kinematics

Velocity-Time Graphs & Acceleration

# Velocity-Time Graphs & Acceleration

Patrick Ford

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Hey, guys, Another type of graph that you'll need to solve motion problems is a velocity time graph. And we can use these graphs to solve things like acceleration. You guys, this is gonna be exactly like how we use position time graphs for velocity. So we're gonna be you re using a lot of the same concepts that we've already discussed with just the variables gonna change. That's really all there is to it. So let's check it out. So with a position time graft, remember that these showed the position in the Y Axis versus time in the X axis. Well, the most obvious difference between these new velocity time graphs is that the Y Axis shows instead the velocity in the Y axis and then the X axis still such time. So really, the only thing that's different is just what the y Axis represents. Now, moving on. We saw that the velocity, which was Delta X over Delta T. Was the slope of our position graph. We'll hear with velocity time graphs. It's the same principle. We're gonna use the slope, except now it's just going to represent the acceleration to the acceleration is the slope off the velocity time graph. Now when we talk about position time graphs, we talked with two different kinds of velocities or slopes. The slope between two points was an average velocity, and the slope of the tangent line at one point was called the Instantaneous Velocity. For example, let's say I wanted the slope between these two points over here, you draw the line, and then this represents the average velocity. So this is my V average. Whereas if I wanted the velocity at a particular points, let's say At T equals four. You have to draw the tangent line, which again was kind of like our best guess here. And this tangent line was just the instantaneous velocity. Well, it's the exact same thing over here. If I want the slope between these two points thes two points, this is gonna represent an average acceleration now. So the only thing is the letter that's different. So this is my average acceleration. And then if I wanted the acceleration at a particular point, then you just draw the tangent line. So this tangent line here is going to represent, or the slope of the tangent line is going to represent the instantaneous acceleration. So the last point is just the steepness of the slopes and what they represent. So in position time graphs. If you had steeper slopes, for instance, if you had slopes that got mawr and more vertical than what that represented is that your velocity was higher in magnitude again, higher just means the number was bigger. Forget about the sign. Soem or vertical meant that the numbers were bigger. Well, it's the same idea here. A steeper slope on a velocity time graph just represents now a higher magnitude acceleration. So the more vertical one line gets, then that means that the acceleration again, the number forget about the sign just gets bigger. All right, guys, so you'll see that we're using the a lot of the same ideas and a lot of the same concepts. So let's just get to an example and see how this works. So we've got the velocity time graph for a moving car over here, and we're going to figure out the acceleration in this first part between t equals 15 and 25. So let's just get to it. So the first thing is that we're solving for an acceleration, but we're using two different times. So which one we're gonna use? Are we gonna use a tangent line? We're gonna use an average. Well, again, this is between two points over here, So this is gonna be an average acceleration. So we're gonna draw the line that connects 15 and 25 which is from here Thio here. And we actually don't have to draw a line because it's already drawn for us. So we just have to figure out the slope of this line so I could make a little triangle like this and I figure out my rise over my run. So this is actually gonna be Delta V over Delta T, not Delta X. And so let's see. So my delta V is gonna be Well, I'm going all the way down to zero. Actually, I'm gonna end up at zero over here, so there's gonna be zero minus and this is gonna be 60. So that means that my Delta V is actually negative. 60. So that's that's negative. 60. And the time is from 15 to 25. So my delta T is 25 minus 15. That's 10. So that means that my average acceleration is negative. 6 m per second squared. So this is my average acceleration. It makes sense because the slope is downwards. So the same sort of downwards being negative. All those same principles still apply here in acceleration. So let's move on to now. The acceleration AT T equals 10. So now we're asked for the acceleration at one particular point, not between two. So that means that we're looking for the instantaneous acceleration when t is equal to 10. So we use the same principle that we did for a position Time graphs. First, we have to figure out the tangent line at this point. So the tangent line here is going to fight trace along this graph, and then instead of following the curvature, I kind of just keep on going in a straight line. So my best guess for this tangent line looks kind of like I don't know, something like this over here. So now this instantaneous velocity is gonna be the slope of this tangent line, or at least my best guess at it. So I still am gonna need a Delta V over Delta t. Remember, I still need to calculate the rise of the run. So I need to points. Except now I'm gonna use these two points over here at the end points. So my final velocity is gonna be 75. My initial velocity is gonna be 30. So my Delta V is gonna be 75 minus 30. So my delta V is 75 minus 30 which is equal to 45. And so I know this is 45 and the Delta T is gonna be from 5 to 15 seconds. So means the my delta T is just 10 seconds. So that means 45/10 is 4.5 m per second squared, and it's positive. So that means that slope of this line, which is the instantaneous acceleration, is about 4.5 m per second. Squared again. Kind of just a best guess. Alright, guys, that's over this one. Let me know if you have any questions

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