WEBVTT
00:00:03.320 --> 00:00:14.570
A particle moves along the 𝑥-axis according to 𝑥 as the function of 𝑡 equals 10 times 𝑡 minus two times 𝑡 squared meters.
00:00:15.520 --> 00:00:19.740
What is the instantaneous velocity at 𝑡 equals two seconds?
00:00:20.270 --> 00:00:24.230
What is the instantaneous velocity at 𝑡 equals three seconds?
00:00:24.850 --> 00:00:28.900
What is the instantaneous speed at 𝑡 equals two seconds?
00:00:29.470 --> 00:00:33.190
What is the instantaneous speed at 𝑡 equals three seconds?
00:00:33.830 --> 00:00:40.710
And what is the average velocity between 𝑡 equals two seconds and 𝑡 equals three seconds?
00:00:42.370 --> 00:00:55.240
The first thing we can notice here about all these different questions is that we’re talking about velocity in the first two, speed in the second two, and then an average velocity in the third.
00:00:56.040 --> 00:01:00.850
And we’re given different values of time at which we wanna solve for these terms.
00:01:02.820 --> 00:01:08.840
Let’s start by writing a little bit of shorthand for each of these five questions.
00:01:09.800 --> 00:01:17.840
We can call the instantaneous velocity at 𝑡 equals two, we’ll write that as 𝑣 of two.
00:01:19.770 --> 00:01:25.880
And likewise, we can write the instantaneous velocity at 𝑡 equals three seconds as 𝑣 of three.
00:01:28.070 --> 00:01:33.880
Now for the instantaneous speed at 𝑡 equals two seconds, let’s call that 𝑠 of two.
00:01:35.000 --> 00:01:40.360
And our instantaneous speed at 𝑡 equals three seconds, let’s call that 𝑠 of three.
00:01:42.800 --> 00:01:45.490
Finally, let’s give a name to our average velocity.
00:01:45.820 --> 00:01:51.500
Let’s call that 𝑣 sub avg, for average.
00:01:53.590 --> 00:01:54.660
Alright, here we go.
00:01:54.660 --> 00:01:58.500
We’ve got our defining equation for motion right here.
00:01:59.000 --> 00:02:04.850
And over on our right-hand side of the screen, we see all the five values we wanna solve for.
00:02:07.020 --> 00:02:15.940
Now the first thing you may notice is that we’re asked for velocity as a function of time, but our function is position as a function of time.
00:02:16.490 --> 00:02:20.460
And we can tell that based on the units written here, units of meters.
00:02:21.160 --> 00:02:29.220
So our first step is to convert this equation to one that tells us what the velocity of this object is, as a function of time.
00:02:30.750 --> 00:02:34.630
To do that, we can recall a definition for velocity.
00:02:35.170 --> 00:02:54.480
Velocity as a function of time is defined as the change in a particle’s position, we’ll use the Greek symbol Δ for change, so the change in the particle’s position as a function of time divided by the change in time elapsed.
00:02:56.440 --> 00:03:11.370
Mathematically, what this means is if we want velocity as a function of time, we can take our position as a function of time equation and take the derivative of that equation with respect to time.
00:03:14.580 --> 00:03:22.420
That means we will take the derivative with respect to time of the equation we’ve been given for position as a function of time.
00:03:24.090 --> 00:03:32.860
In other words, velocity is a function of time is equal to the time derivative of 10𝑡 minus two 𝑡 squared.
00:03:35.510 --> 00:03:49.790
Taking the derivative of that function with respect to time, we end up with an equation for velocity as a function of time of 10 minus four 𝑡 meters per second.
00:03:53.150 --> 00:04:00.850
This is the equation we’ll use to solve for the five different values that our question asks for.
00:04:03.850 --> 00:04:09.840
Let’s start off by solving for velocity when time equals two seconds.
00:04:11.000 --> 00:04:22.190
So if we plug in two for 𝑡 everywhere we see it in this equation, we find- we get 10 minus four times two meters per second.
00:04:24.470 --> 00:04:29.690
That’s 10 minus eight which equals two meters per second.
00:04:33.370 --> 00:04:39.170
That is our instantaneous velocity when time equals two seconds.
00:04:41.140 --> 00:04:46.810
We’ve now solved for 𝑣 when 𝑡 equals two seconds, so we can cross that off our list.
00:04:48.160 --> 00:04:52.080
Now we move on to 𝑣 when 𝑡 equals three seconds.
00:04:52.980 --> 00:05:02.700
Plugging in three every time 𝑡 appears in this equation, we find 10 minus four times three meters per second.
00:05:05.170 --> 00:05:11.780
This equals 10 minus 12 which is negative two meters per second.
00:05:15.210 --> 00:05:20.830
That’s our instantaneous velocity when time equals three seconds.
00:05:23.660 --> 00:05:31.030
Now we get to an interesting point because we’re no longer solving for instantaneous velocity, but we’re solving for instantaneous speed.
00:05:31.810 --> 00:05:35.920
Now remember that speed is a scalar quantity.
00:05:36.750 --> 00:05:46.800
So when it comes to our equation, speed as a function of time will equal velocity as a function of time within absolute value bars.
00:05:47.300 --> 00:05:53.650
Speed is the magnitude of velocity working off of this equation for velocity as a function of time.
00:05:54.950 --> 00:06:11.180
What this means is that when we wanna solve for speed when 𝑡 equals two seconds, all we need to do is put absolute value bars around the same equation for velocity when 𝑡 equals two seconds.
00:06:16.390 --> 00:06:25.560
We can see that when 𝑡 equals two seconds, 10 minus eight is already a positive number, so we need not worry about the absolute value bars.
00:06:25.830 --> 00:06:31.810
And speed, when time equals two seconds, is equal to two meters per second.
00:06:35.350 --> 00:06:41.520
So our instantaneous speed at two seconds is equal to our instantaneous velocity at two seconds.
00:06:43.600 --> 00:06:48.850
Now let’s move on to the instantaneous speed when time equals three seconds.
00:06:49.810 --> 00:07:02.990
So we again use our absolute value bars, 10 minus four times three, close up with our absolute value bars, and complete with our units, meters per second.
00:07:04.500 --> 00:07:11.030
Now as you look into that equation, you see that we’ve got 10 minus 12 or negative two.
00:07:11.500 --> 00:07:16.960
But because of our absolute value bars, that flips the negative to a positive.
00:07:17.240 --> 00:07:22.820
And we wind up with an answer of positive two meters per second.
00:07:24.980 --> 00:07:32.710
So in this case, our instantaneous speed at three seconds is not equal to our instantaneous velocity at three seconds.
00:07:33.070 --> 00:07:35.450
Thanks again to our absolute value bars.
00:07:40.430 --> 00:07:52.810
Now we’re left with just one more thing to solve for, 𝑣 sub avg, the average velocity of the particle between time equals two seconds and time equals three seconds.
00:07:53.930 --> 00:07:58.820
We can start by defining what average velocity is.
00:07:58.850 --> 00:08:16.500
Mathematically, the average velocity between two points in time is equal to the velocity at that initial point in time plus the velocity at the final point in time divided by the change in time.
00:08:19.800 --> 00:08:28.910
When we write this out for our own given values, we see we’ve already solved for 𝑣 of two, that’s two meters per second.
00:08:29.580 --> 00:08:36.180
And we’ve also already solved for 𝑣 of three, that’s negative two meters per second.
00:08:38.750 --> 00:08:43.670
Now the time involved is between 𝑡 equals two seconds and three seconds.
00:08:44.030 --> 00:08:47.200
So our change in time is one second.
00:08:49.430 --> 00:08:52.660
Now take a look at the numerator in this equation.
00:08:52.960 --> 00:08:58.320
We have positive two meters per second plus minus two meters per second.
00:08:59.010 --> 00:09:13.150
So what we get is: zero meters per second is our average velocity between times 𝑡 equals two seconds and 𝑡 equals three seconds.
00:09:15.110 --> 00:09:18.540
This concludes all the values we wanted to solve for.