00:01
So they walk us through some steps for us driving some car 60 miles per hour down a road, and we're trying to come to a complete stop within 242 feet.
00:13
So we want to determine what constant deceleration we're going to need to ensure this.
00:18
So the first thing they do is they give us this differential equation here, or this initial value problem.
00:28
So remember, s prime is telling us the derivative.
00:32
Of our displacement.
00:34
So that's really just velocity.
00:35
So that's where the 88 is coming from, because that's our initial velocity.
00:39
And our initial distance, before we sum on our break, should just be zero.
00:43
So that's where the s of 0 is equal to zero comes from.
00:46
So i tell us to solve this differential equation, though.
00:48
All right, so let's first go ahead and integrate our second derivative here.
00:55
So integrating that should give us s prime on the left, and integrating a constant.
01:00
So if we integrate this constant here, well we just multiply by our variable.
01:06
So it would be negative kt plus c.
01:10
Well, we know that our initial velocity was 88.
01:16
So let's go ahead and plug that in.
01:17
So it's going to be 88 is equal to 0 plus c.
01:20
So that gives us c is equal to 88.
01:24
So we can go ahead and plug that in now.
01:27
And we'll get s prime is equal to negative kt plus 88.
01:34
Now we can go ahead to integrate this.
01:36
One more time to get our displacement function.
01:39
So just integrate each side again with respect to t.
01:43
So that's going to give us that s is equal to.
01:47
Well, integrating t, we'll just use power rule.
01:52
So that would be negative k over 2, t squared, and then plus 88t plus our constant c.
02:02
So again, we can go ahead and use our initial condition that at the start, we haven't moved it all so that's going to be 0.
02:09
So we have 0 is equal to 0 plus 0 is equal to c.
02:13
So that tells us c is 0...