https://youtu.be/gG2p8FdryRA
PHYS 1101: Lecture Seven, Part Four
In this section, let me walk you through a couple of trajectory examples. Here’s a very key point. It’s highlighted in red here for you. We would need to be able to make the distinction between when a certain initial velocity and a certain acceleration is going to lead to one dimensional motion or two, meaning, when is our motion going to just follow a straight line, or when will this object actually curve, follow a curve trajectory.
Here’s the rule of thumb. The only time you will get a curve trajectory is when you have an initial velocity, you have an acceleration, and these two vectors are not parallel. Any other scenario, and the object’s going to go along in a straight line. It may slow down along that straight line. It may speed up. It might even slow down, turn around, and trace back on itself, but it will be along the straight line. You have to have initial velocity and an acceleration that are not parallel.
Here’s three examples for you. Initial velocity. I have different orientations, and I have different examples of an acceleration vector. Notice in all cases, they’re not parallel. They’re off at some angle.
Let’s quickly walk through and show you how you would predict the real trajectory, the real curved path that each of these scenarios would follow. I’m going to start with this first one. Copy it down here for you. Okay. Given this initial velocity, if you like, let me put a v0 here to make it clear. With this initial velocity and this a, what motion do I end up with? Let me grab my black marker and know… Well, here’s the first two points on my trajectory. What happens next?
Well, next I know that v0 plus delta-v gets me my new velocity. But, to clearly put it in the right spot, I’m actually going to copy v0 over, just like we did above, to the next point. I’m going to add delta-v to then sketch tail to tip sum, my next velocity. If that’s v0, let me call this v1. Then, I’ll just start numbering them.
Okay. Let me erase this because I just used that as my intermediate tool to get the next trajectory. Now, let me repeat it to get another point. Here’s this point. Now, I’m going to copy this vector. I’m going to move that to the next point. This is what would happen if I had no acceleration, but I have acceleration.
So again, I have to do, to this velocity, I have to add this delta-v in order to get my actual next velocity. I’m going to erase this again because I used that just for construction purposes. Here’s my next point.
So, you can imagine continuing this to see that with this continuous downward acceleration, always adding a delta-v of the same amount, I cause this kind of curved trajectory. This initial velocity, then this one, then this one. You can picture how it would continue on. This object starts out initially going straight, but then it gradually starts curving down because of this acceleration.
Okay. The next trajectory. Let’s do this one real quick. I’ll just do a couple of points again for you. Here’s my v0, and then let me just erase my a, so it doesn’t muck up my drawing because it’s clear what that is. My red arrows are always acceleration. So, here I’m going to repeat v0 at the next point. Oops. To that, I’m going to slide my acceleration vector over. And now, I know that my next velocity goes from the initial tail to the final tip. I’m going to grab this and move it down a little bit.
Bring this over here and tidy it up. I’m going to need that acceleration vector. I’m going to go in and erase this artificial next velocity I drew, just for the construction. I’m going to erase this acceleration. Let me put my points in here. I’m going to repeat it one more time.
So now, this vector, I’m sliding to the next point. But, I know rather than continuing on straight, I have to add this vector to get my new velocity vector, from the initial tail to final tip. Let me go in again. Sorry about that. I’m trying to grab. There we go. The arrow for my acceleration. And of course, erase. I’m erasing those artificial or intermediate vectors I drew just for my construction. So, this is the trajectory that this object would follow, given this initial velocity and this acceleration.
So, notice the more opposing or away from parallel these two vectors are, the more dramatic my curve. Here, I had 90 degrees, and that causes the object to turn. Here, the two vectors are in a similar direction, similar magnitude, so I get only a very gradual arcing. This one’s going to be interesting because this acceleration is quite opposite or very opposite direction tendency compared to this.
Let’s grab these and see what we’ll end up with. Maybe I’m going to speed this up here and just make a couple of copies of my acceleration vector. Okay. Here, I’ve labeled v0. I’ve got a couple of copies of my acceleration vector. This will be quicker for me to construct.
Step one is to make a copy of that previous vector and just stack it on top of the existing. But now, I know that that’s to get to my new velocity. I have to add this acceleration vector. So, here. Tail to tip, this is my next v. Now, let me erase these, and I’m going to do it again.
To this velocity, rather than it being the same, continuing on in the same direction, I have to add my acceleration. My next velocity then goes from this tail to this tip. I’m going to erase these two that I used to construct it.
Let’s do it one more time, just because this motion is more interesting. This next velocity or to the last velocity, I’m going to repeat it. But, instead of continuing on, I know that I actually have to add delta-v to get initial tail to the final tip, to get that vector.
So, that’s a pretty good representation of what happens here. Probably done enough points now. So, with this acceleration and this kind of initial velocity vector, I end up with this motion that follows this gradual arc.
So, there’s three different scenarios to get a feeling for how do you construct these trajectories and what kind of acceleration will lead to a particular path, given the direction the object initially headed. There’s three scenarios for you.
So, notice with the initial velocity and the acceleration, that the more nearly parallel and pointing in the same direction they are, the more gradual and straight the trajectory. The more opposite the tendencies are, the more hooked or curved the trajectory is going to be. If these were actually parallel but pointing in opposite directions, that’s when I end up with the motion that comes to a stop and swings back on itself. And then, this scenario is somewhere in between. These two happen to be at 90 degrees, and consequently, end up with kind of just a gradual arc down.