Draw position, velocity, and acceleration graphs for the ball shown in FIGURE P2.44. See Problem 43 for more information.

Video transcript

Hey, everyone in this problem, we have a virtuous trolley following a straight path as shown in the figure below. And we're asked to represent its motion using position versus time velocity versus time and acceleration versus time graphs. OK. We're told to assume that the position S is equal to zero at the point where the trolley is shown. We have our trolley shown it's on this ver or sorry horizontal section. We have that the initial velocity is greater than zero. And this trolley is moving to the right, it's gonna move to the right along this horizontal section. It's gonna encounter this incline. So it's gonna go downwards on the incline and then hit another horizontal section at the bottom of that incline. OK. All right. So we're gonna break this up into three segments and we have kind of three segments of the motion here. So we have this first horizontal section, we have our incline section and then the third horizontal section and we're gonna start with our acceleration versus time graph. OK? I find it a little bit easier to start with the acceleration graph and work backwards to the velocity. And then the position just because the position is always a little bit more involved. OK? And you'll see that in just a minute. So we're gonna start, we're gonna draw out our axis and the first one, we have time on the X axis, acceleration on the Y axis and we're gonna draw our three segments. OK? So we're gonna break this up into our three segments so that we can compare, let me just make that more even so that we can compare this to our. Now, one thing that's important to note here is that this is a frictionless troll. So this horizontal section that it's on right now, it's gonna have no acceleration. OK. Why is that? Well, there are no forces acting on it in the direction of motion. OK. The trolley is moving in the horizontal direction. There's no friction acting in the horizontal direction. There's nothing else. OK. So we have no acceleration on this trolley. So we're gonna draw just a zero acceleration for that entire signal. OK? Just a nice horizontal line at zero. Now is Charlie's gonna reach the downward in England at this point and we still have no horizontal forces acting, but we have some motion in the vertical direction because the trolleys on the incline and the force of gravity is going to act in that direction to accelerate the block and the force of gravity is acting in the same direction as the vertical motion. We get this positive acceleration. So for the second segment, we're gonna have some positive acceleration. We don't know the value of that. We aren't giving any, any information about the angle of this incline or anything like that. But we are gonna have some positive constant acceleration throughout that second segment. Then we're gonna come to the third segment and we're back to a horizontal section. We have no friction acting, there's gonna be no forces acting in that direction of motion. And again, we get a zero acceleration. OK. So the acceleration versus time curve is just a simple curve, just kind of one line nonzero throughout the second segment. Hey, this horizontal line. All right. So now that our acceleration is done, we're gonna move to the velocity and we're gonna draw out our axes again and make sure we can see it all. All right. And then we're gonna write our segments with the time on the X axis and the velocity on the Y axis. Now, in this case, we're told that the initial velocity is nonzero, right? It's positive. So we're gonna pick some initial point for velocity and being not and we don't know the exact value, but we do know it's positive. So we're just gonna choose a point there along our first segment. Hey, this is a horizontal segment. We're, we've already determined there's no acceleration. If there's no acceleration, then the velocity is going to remain constant. OK? So for this first segment, we're gonna start at our initial velocity V not s and we're just gonna remain at that velocity for the entire segment. OK. Just this positive horizontal line. Now, we can always double check when we're doing these, that we're on the right track. I recall that there's a relationship between the velocity and the acceleration. The acceleration is the derivative of the velocity. If we were to look at our velocity curve here, we have this constant positive. Why if we take the school uh the derivative of it, what's the derivative of a positive constant? Well, it's just zero. OK. And that's exactly what we have for the acceleration at that point. OK. So we can always double check that the relationship makes sense. Moving to the next one, the next segment we have this downhill incline. We know that we have a positive acceleration that is constant. We have a positive acceleration that means that the trolley is gonna be speeding up, it is going downhill, gravity is gonna help speed it up. And so we're gonna have this straight line with a constant slope for the second segment of our velocity, that velocity is increasing. And again, we can look at the relationship if we were to take the derivative of a line with a positive lope, it would be a positive constant value, which is exactly what we have for the acceleration. All right. Now, once we get to that bottom incline or sorry, the bottom horizontal section, we're back to having no acceleration. So the speed is gonna remain constant, the velocity is gonna remain constant and that velocity is gonna remain at what it was at the bottom of the incline. And so we have a straight horizontal line coming from the end of segment, we have two segments with constant positive velocities. The third segment has a higher velocity than that initial one and the segment in between segment two has this linearly increasing velocity. All right. So two of the three graphs done now we're gonna move to our final graph and that is going to be the position versus time. Hey, and I'm gonna draw it on the side on the left here just so that we can keep our diagram and all of our graphs up at the same time if we dry out our position versus time. So we have our time t on the X axis position s on the Y axis and we have our three segments. Now, the first segment, we have a constant positive philosophy, just like the relationship between the velocity and acceleration. We have a relationship between the position and the velocity. And again, the velocity is going to be the derivative of the position. So we're, if we have a constant steady speed, our position is going to change linearly, it's gonna change by the same amount at every time. And so we are going to have a line with a positive slope for S in the first segment, we know that S starts at zero. OK. And so we're gonna have some linear relationship. OK. And again, we can double check, we take the derivative of this line with a positive slope, we get a positive constant, which is what we have for our velocity and segment. OK. So always double checking that the relationship makes sense. Now the second segment, this gets a little bit trickier. OK. Our speed here, our velocity is increasing. We're going faster and faster and faster. What that means is that for every time point we're gonna be traveling further and further, right? We're no longer gonna have a line. We're gonna have a quadratic equation that's increasing upwards to indicate that we are covering more distance over the same amount of time. OK. So we start with our speed that we had and we are going to have some quadratic like curve that increases upwards, right? All right. OK. Taking the derivative of quadratic, we get a linear function which is what we have for segment two of our velocity. Now, this last segment, we are back to this steady speed. OK? This constant velocity, but that velocity is higher than the initial velocity. And so the slope is going to be steeper. And I'm kind of cutting off into our diagram a little bit just to make it clear that this is another straight line. OK. So we have this linear section with a steeper slope than what we had before. All right. So now we have our position, our velocity and our acceleration curves for this trolley and that's it. Thanks everyone for watching. I hope this video helped see you in the next one.