00:01
Hi there.
00:02
So for this problem, we need to calculate how long it will take for the second jumper to reach the same velocity that the first jumper reach in 10 seconds.
00:15
So we are giving some information for the first and the second jumper.
00:21
The first thing that we are going to do is to draw all the forces that are acting on a free fall body like this.
00:31
That is not only experience the weight force, but also a drag force that opposed to the direction of emotion, in this case, downward.
00:45
So, and after that, we are going to obtain a differential equation for the velocity.
00:53
We'll solve this equation and solve for the velocity, obtain the velocity for the first jumper.
01:01
And then we substitute that for the time of the second jumper.
01:08
And first, as i said, i'm gonna draw in here an ad set.
01:15
I'm gonna assume that the downward motion is, and the positive, so i'm gonna put it here positive, and this is the negative, sorry.
01:30
So if we have, they're gonna assume that this point is a jumper.
01:36
The only forces that are opting on a jumper are the drag force that are going to call f of d, and the weight.
01:49
We know that the weight is defined as the mass times the acceleration due to gravity.
01:54
And we know that the drag force is defined as the product between the coefficient of drag times the velocity.
02:05
So applying newton's second law, we will obtain that the force is the sum of all of all of these forces.
02:15
So because we said that downward is the positive direction, so we will set that the weight is positive minus the drag force that we have defined, that we know it is defined as this.
02:31
And this should be equal to the mass times the acceleration because the system is moving, jumper is moving downwards.
02:42
So with this, we can remember that the acceleration can be rated as the derivative of the velocity with time.
02:57
So we can substitute that in here and also divide all of this by the mass n.
03:06
So we will have something like this.
03:09
G minus k over n times b.
03:14
It's equal to.
03:18
So we have obtained a differential equation, and we need to solve it.
03:25
So we are going to pass this differential of t to the left side, and this expression to, the right side because the right hand side turn is the one that depends on the velocity p.
03:45
So doing that we will obtain something like this.
03:59
And we just need to integrate this.
04:05
The first integral, the right hand one, is just simply the time plus a constant, that is the integral constant.
04:18
And from the other part, we're going to make a change in variables to make an easier integral.
04:27
We are going to set that u is equal to g minus km times b, so that the differential in the u, it is minus k over m, dv, because that's the only variable in here.
04:44
The other are just constants, and we can solve for the differentiation in velocity, so that is this.
05:00
So we can substitute all of this in here, and we will have something more simplified.
05:09
And we will have a turn that can be taken out of the integral, because it is a constant, this turn in here.
05:18
Is a constant and we can just take it out of the integral and we will just simply have this integral and we know the result of that integral we know that it is the neparian logarithm of you and we know what is the expression for you so we just substitute that g minus k of b.
05:56
Now with this, the first thing that we need to obtain is the velocity for the first jumper, because we need that velocity to solve for the time in the second jumper.
06:11
So we need to solve for b.
06:13
And we first pass this turn to the other side.
06:19
We'll have something like k over m times theima plus c is equal to the logarithm of g minus k over m times b.
06:36
Now we apply the exponential function because that is the inverse function of the neparian logarithm.
06:50
And we know that the exponential of the sum is the product of the exponential.
06:58
Of each term, so that's what i'm doing in here...