00:01
In this problem you're given the position vector and then you're asked about velocity, acceleration, the velocity at a certain time, the acceleration, but then also about the average.
00:10
I know that's your particular question, but let's go through everything just quickly anyway.
00:16
V of t, dr of t, dt, time derivative of position with respect to time.
00:24
So, and we know that derivative of t to the n with respect to time, n t n minus 1.
00:30
T x, it's exactly the same.
00:34
So we get here 6 .0 t i -hat minus 21 .0 t squared j -hat minus 5 .0 minus 2 t minus 3 k -hat.
00:51
This is meters per second now.
00:53
And cleaning it up, 6 .0 t i -hat minus 21 .0 t squared j -hat plus 10 t minus 3 k -hat meters per second.
01:07
So that's the velocity.
01:11
Now they want the acceleration.
01:16
And that is dv dt or likewise the second derivative of r.
01:23
Doesn't matter, you got the first derivative here, take the second, take the derivative again.
01:28
Alright, so we get 6 .0 i -hat minus 42 .0 t j -hat plus 10 times minus 3 times t to minus 4 k -hat meters per second squared.
01:48
Cleaning this up a little bit, 6 .0 i -hat.
01:52
You might say why, i only asked about the last part.
01:55
Well, i need, there is something i need to discuss about that part with the acceleration.
02:01
So that's why i needed this here.
02:03
6 .0 i -hat plus 42 .0 t j -hat minus 30 .0 t to minus 4 k -hat meters per second squared.
02:16
So that's the acceleration.
02:19
The next part, they want the velocity at 2 seconds.
02:27
You're going to see me doing that with the average anyway.
02:30
You just put in the 2 seconds, put 2 in for t into this formula here.
02:36
And just calculate.
02:37
That gives you, that gives you your answer.
02:41
D wants the speed, v at 1 .0 seconds.
02:50
This is just the pythagorean theorem.
02:52
You got 3 components.
02:54
You're going to have vx squared at the 1 .0 seconds, vy squared 1 .0 seconds plus vz squared 1 .0 seconds.
03:10
That's it.
03:12
You're just doing the pythagorean theorem twice.
03:17
You're really looking, you're getting in the xy plane and then you got to add the z aspect and do the pythagorean theorem with that.
03:24
So whatever this is becomes part of that pythagorean theorem.
03:31
You know when you're thinking about it, you have a vector like this.
03:41
This is the vx squared plus vy squared.
03:44
You know this is x, y, and z.
03:48
But then you have to get this.
03:50
But this is one side.
03:51
Remember this is a 90 degree angle, it's straight up.
03:56
You have one side, you have the z side, and then you get the length of the full vector at that point.
04:04
That's all that's going on here.
04:13
And this is vz.
04:15
So when you square this, when you square this with the pythagorean theorem, you're going to get the vx squared, the vy squared, and the vz squared.
04:28
That's all there is to it.
04:31
And you can punch in the numbers there too.
04:33
You have everything here too.
04:35
So you put in 1 second.
04:37
And i also asked for 3 seconds, so you put in 3 in the other one.
04:40
That's, i see that you did not ask about that, but still it's good to go over just in case.
04:49
E, the average.
04:53
Now you wrote this.
04:58
You have 1 .0, you have 2 .0 over 2.
05:05
That does not apply here.
05:09
Only if a is equal to a constant.
05:17
Technically, if you're looking at components, ax would be a constant, ay would be a constant, az would be a constant, because we're looking, really that represents 3 equations.
05:26
So you could have, this could work for 1 or 2 of those, but not necessarily all 3.
05:33
But let's just deal with it as a full vector.
05:36
You can see the accelerations.
05:40
Yeah, so if you wanted to get the average for the x components, yeah that's constant.
05:55
But neither the y and the z are not.
05:59
Acceleration is not constant.
06:02
So that does not apply.
06:05
Let me give you an example.
06:08
Say, maybe think of these as test scores, it doesn't matter.
06:13
Or you want to think of them as speeds.
06:15
Let's say step of 1.
06:18
So my acceleration is 1 meters per second squared.
06:21
So i'm going from 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100.
06:36
And you were doing average, where you added all those up, divided by 11, because there's 11 things, you get 95.
06:44
Okay, now you try doing this.
06:50
The first and the last divided by 2, really that's what you're looking at, the beginning of the interval, end of the interval.
06:55
90 plus 100 over 2 is 95.
07:01
Okay, all's good.
07:03
But that's an even step.
07:07
Now let's say step increases.
07:13
So the first step is 1.
07:16
The next step is 3.
07:18
So your acceleration is going from 1 meters per second squared to 2, to 3, to 4...