00:01
Hi there, so for this problem, we are given the following differential equation that will be that the rate of change of the velocity with respect to time is equal to 9 .8 and then this minus b divided by five.
00:21
And we are also given that the initial velocity that will be the velocity at zero is equal to zero.
00:27
So for part a of this problem, we are asked to find a time that must elapse for the object to reach 90 % of its limited velocity.
00:36
First of all, we need to solve this differential equation and for that we can do this and something as follows.
00:44
We can take out one divided by five, sorry, minus one divided by five as a common term in here.
00:53
When we would do that, we need to multiply everything by minus five.
00:57
So we obtain b and 9 .5 times five is 49.
01:04
In this case is negative because we take out minus one divided by five.
01:08
Once we have in this, we just need to, well, we can just move this to the other side to divide.
01:14
We are separating the variables just like this.
01:18
And then for the other side, we will have minus one divided by five times the differential in time.
01:25
Now, once we have this, what we can do is to integrate both sides of this.
01:31
Now for the left side, we obtain just simply the neperian logarithm of b minus 49.
01:39
And for the right side, we obtain minus one times the time plus a constant of integration that we're gonna label as c.
01:47
Now to get rid of the neperian logarithm, we are going to apply the exponential in both sides of this.
01:55
So we will have that b, solving for b, that will give us that this is just 49 plus c times the exponential of minus the time divided by five.
02:06
Once we have this, we need to use the initial condition that the velocity at zero is zero.
02:12
So if we evaluate this at zero, remember that the exponential of zero is one.
02:22
So we obtain that c is just minus 49 by using that condition.
02:26
So with that said, the velocity now becomes just simply 49 times one minus the exponential of minus the time divided by five.
02:35
So once we have this, now we are asked to find the time that must elapse over the other to reach its limiting velocity.
02:43
Now, first of all, let's find the limiting velocity.
02:45
We obtain that by doing the limit when the time tends to infinity of the expression that we obtained.
02:52
And here, remember that when the exponential tends to minus infinity, that tends to one.
02:58
So then that tends to zero, sorry.
03:01
So we will have one minus zero that will give us that the limiting velocity is 49.
03:06
Now, in this case for part a, we are asked about the object to reach 90 % of its limited velocity.
03:12
So we want when the velocity of this will be 49 times 0 .96.
03:19
And we set this equal to the expression that we are given for this.
03:23
Now we can start simplifying in here this 49 with this 49.
03:28
Then once we have this, what we can do is to pass this to this side and this to the other side to subtract to the one.
03:36
So we will have the exponential of minus the time divided by five is equal to one minus 0 .96...