00:01
All right, let's say we have some function v of t equal to 10 minus 4t, which describes the velocity of a particle on a line on the time interval 0 to 0 .5.
00:12
And let's say v is in meters per second, so t the time is in seconds, and the distance is in meters.
00:19
So we want to find two things.
00:21
One is the displacement as a function of time, which we'll call delta x of t.
00:26
And the other thing we want to find is the total distance on the time interval.
00:31
Okay, so first let's start with one, right? so v is equal to dx d t, and we say delta x of t is x of t minus x of zero, right? so using our knowledge of calculus, we know that x of t minus x of zero is the integral from zero to t of dt of t.
00:48
Then we just plug in, right, 10 minus 4t.
00:51
What's the integral of 10? 10t? what's the integral of minus 4t? what's minus 2t squared? and we do it from 0 to t so that's just 10 t minus 2 t squared and we're done with that part so that's the displacement function time now they want the total distance from 0 to 0 .5 right so now we integrate from 0 to 0 .5 d t absolute value of vt because it's distance distance you always do absolute value however we're in a we're in a bit of luck here because interval is from 0 to 0 .5.
01:35
So 10 minus 4 is always positive on that interval...