00:01
Okay, so a particle moving in a straight line is given by the acceleration function, a of t equals, let's see, 2t minus 4.
00:13
And we're also given that its position at time equals 0 is negative 5, and that 2 seconds of travel gives it a velocity, so v of 2 equals negative 1.
00:29
So we want to find the, at the very end, we want to find the position function.
00:36
And so first we need to find the velocity function.
00:40
And that is given by d of v2 over dt equals acceleration.
00:50
And we know that just because acceleration is the time derivative of velocity.
00:57
And so d vt over d t equals 2 t minus 4.
01:10
Now, while mathematicians don't like this, we just multiply both sides by d t, and then we get d t over here.
01:20
So d of vt equals 2t minus 4 d t.
01:29
We take the integral of both sides we get v of t equals t squared minus 4 t plus c and we're going to use the condition that v of 2 equals negative 1 to get this c so b of 2 equals negative 1 equals negative 1 equals 2 minus 4 times 2 plus c.
02:09
And negative 1 equals 4 minus 8 plus c.
02:17
Over here.
02:19
So we get negative 5 equals negative 8 plus c...