00:01
We want to know the velocity of the object, given that its acceleration is given by the following function of time, and the object has an initial velocity of zero in that it starts from rest.
00:13
So we use integration with respect to time to find the velocity.
00:19
So we'll go ahead and get started here.
00:22
A, not, and b are functions, or rather constants.
00:26
And it's important to know their units, because this can be helpful in checking results.
00:31
So notice that b multiplies t and the argument of the exponential function needs to be dimensionless.
00:38
So b has the units of inverse time or basically inverse seconds.
00:44
A not multiplies a dimensionless quantity, which is the function itself.
00:50
So since this is an acceleration, a not should have the units of meters per second squared.
00:57
So we'll go ahead and just note the units for these constants.
01:00
Okay, so we'll go ahead and determine the acceleration, or rather the velocity as a function of time.
01:10
And we'll also use this initial condition in getting the constant of integration.
01:15
So we'll integrate here.
01:22
So in taking the integral of an exponential, the first thing we can do is we can pull this constant a -0 outside of the integral and focus entirely on the exponential function itself.
01:35
In integrating a constant inside of an exponential basically goes in the denominator.
01:44
We divide by the constant itself.
01:46
So we have a not divided by negative b, and we just have the exponential function itself.
01:55
We also have a constant of integration right here, but we know that at times zero, this velocity must be zero.
02:08
So we will set this equal to zero, and we will plug in the value t equal to zero in the function.
02:17
So e to the minus zero is this quantity is going to be equal to one, and this constant can be determined.
02:24
So now this gives us 0 equals negative a not over b plus the constant itself.
02:42
And we can add, we can say zero over here, and we can say minus a not over b plus a0 over b.
02:53
These will cancel.
02:56
We can add a not over b to this side also.
02:59
And this will basically be the constant of integration.
03:04
I'm going to make some room, and the constant of integration is just a -0 over b.
03:14
So the velocity is the function of time is equal to coming back up a little so we can see what we had before.
03:29
Negative a knot over b, e to the minus b, t plus a not over b.
03:40
And we can rewrite this to factor a little bit and simplify.
03:45
So we can pull a not over b out, and then we'll have negative 1 times e to the minus b t plus 1.
03:59
And then the two terms in parentheses can be rearranged, and we have one minus e to the minus bt.
04:12
So that is the velocity as a function of time.
04:18
Now, we want to know what happens as time goes to infinity.
04:23
How does the velocity behave as a function of time in the limit as t goes to infinity? so what this means is we take the limit as t goes to infinity.
04:34
To infinity of the quantity that we just obtained.
04:44
Now, the exponential term will actually just go to zero.
04:52
So we end up with a constant.
04:55
We end up with a -0 over b.
04:58
So this term, this one will just stay as one.
05:02
This term will go to zero.
05:04
And so we have one minus zero, and we end up with a -0 over b.
05:10
And so the velocity approaches a constant value of a not over b.
05:15
So we have a finite velocity.
05:17
We have finite velocity of a0 over b as t goes to infinity.
05:38
Next, one thing that we want is the position as a function of time.
05:44
So we take a quick look at the velocity and we say, okay, well, we can integrate this.
05:50
We say position as a function of time is the time integral of the time.
05:55
The velocity.
05:56
So we just substitute in our function that we obtained.
06:06
And we don't necessarily know what position it started at.
06:10
We could impose a condition that it started at the origin, but we don't have to.
06:15
For our purposes, it won't matter.
06:17
I'll just say that it started at time zero, the initial position was some general value of x not.
06:25
And that will suffice because for the of this problem it'll be sufficient in understanding the behavior of this function that we're about to get for the position so we'll go ahead and we'll carry out the integration for this and we get the following next we get we pull out the constant factor on the left and we have in parentheses here or inside the integral we get t now we have like before we put negative b on the bottom is basically dividing the exponential function.
07:14
We have a minus sign from the fact that we were subtracting the exponential and this comes out to a knot over b times t plus e to the minus b t over b but one thing i neglected to do is include the constant of integration.
07:46
So let me rewrite this and we'll just continue on the next line...