00:01
Okay, so the acceleration is 2t plus 2, and we know that v of 0 is equal to minus 15.
00:18
And we want t between 0 and 5.
00:22
And we want to know what is the distance traveled.
00:28
Okay, the way to do this is to first find the velocity.
00:32
So the velocity is the integral of the acceleration.
00:38
So we integrate 2t plus 2, which is just t squared plus 2t, and then plus a constant.
00:47
And to find the constant, we use this condition, the v of 0 equals minus 15.
00:51
So if i plug in 0, this has to equal minus 15.
00:56
But if i plug it into this equation, this is just a constant.
00:59
So this tells me that v of t is equal to t squared plus 2t minus 15.
01:07
Okay and we're also going to factor this just to make it easier to read so we can read this as t plus five times t minus three if you factor it okay so what do we see uh so t is between zero and five so let's look at the the velocity where the velocity is positive and negative to find a total distance traveled you need to calculate the absolute value of the velocity.
01:46
So for total distance, we need to know the integral of the absolute value of the velocity from 0 to 5.
02:00
So to find this, let's figure out where it's positive or negative.
02:03
So we see that it crosses 0 at 3.
02:10
So i guess i'll do this later again.
02:15
So let's first do this.
02:17
So it crosses at 3, 0 and then if it's below 3 so if it's let's say 1 then this is negative and then if it's above 3 let's say like 4 or 5 then it's positive okay so to calculate the absolute value of the velocity d t this you can split this up between 0 and 3 you take negative of the velocity since it's negative there and then between 3 and 5, you take the velocity.
03:01
Okay, so now let's actually just use the formulas and compute.
03:07
So if i integrate the velocity, just by power rule, i get t cubed or negative the velocity.
03:16
So i can bring the negative sign out.
03:18
So i'll bring negative sign here.
03:21
So i get t cubed over 3 plus t squared minus 15t...