Current time:0:00Total duration:5:07
0 energy points
Ready to check your understanding?Practice this concept
As an example of finding a mean value, we find an average acceleration.
Video transcript
Let's say that we have a particle that's travelling in one dimension and it's positioned as a function of time is given as t to the third power plus 2 over t squared. What I would like you to do, is pause this video. And figure out what the average acceleration is of this particle over the interval, the closed interval, from t is equal to 1 to t is equal to 2. What is this, what is this going to be equal to? So assuming you've given a go at it, and the first thing you might have realized is that we're trying to take the average value of a function that we don't know explicitly yet. We know the position function, but not the acceleration function. But luckily, we also know that the acceleration function is the derivative with respect to time of the velocity function which is the derivative with respect to time of the position function. So the acceleration function is the second derivative of this. And the we have to find its average value over this interval. So let's do that. Let's take the derivative of this twice. But before we do it, let me just even rewrite this. So it's gonna be a little bit easier to differentiate it. So if we just take each of these two terms in numerator and divide them by t-squared. We're gonna get t to 3rd divided by t-squared is just t. And then 2 divided by t square, we can write that as plus 2 t to the negative 2 power. And now lets, lets take the derivative. So the velocity function as a, as a, the velocity as a function of time. The derivative of this with respect to time, so it's going to be derivative of t with respect to t is 1. Derivative of 2 t to the negative 2, let's see negative 2 times positive 2 is negative 4 t to the, and we just decremented the exponent here. t to the negative, negative 3 power. And now to find the acceleration as a function of time, we just find and take the derivative of this with respect of time. So acceleration as a function of time is equal to, actually since I've already used that color for the average, let me do a different color now. So acceleration as a function of time is just the derivative of this with respect to t. So derivative of a constant with respect to time, well it's not changing, so it's zero. And then over here negative 3 times negative 4 is positive 12 times t to the let's decrement that exponent to the negative 4 power. Now to find the average value, all we have to do now, average, average value is essentially take the definite integral of this. Over the interval And divide that by the width of the interval. So or we could say, we could take, we could divide by the width of the interval, one over two minus one. And this all simplifies to one. Times the definite integral over the interval. So one to two, of a of t, which is, so that's gonna be 12 t to the negative four power dt. So what is this simplified to? Once again this is 1 over 1, that's just going to be one. We take the anti derivative of this, we actually, well let me just, this is going to be equal to the anti derivative of this, so we're gonna go t to the negative 3 power but then we divide by negative 3, so n anti derivative of this is going to be if we don't take the. Well, an anti-derivative is going to be negative 4, t to the negative 3 power, and we saw that over here, obviously, if you were really just taking an indefinite role, we would have to put some constant here, but in the definite rule, even if we put a constant here, it would get can, if we, if we, assuming the same constant, it would get cancelled out when you actually do the calculation. But the anti-derivative of this, we increment the exponent. And then we divide by that new exponent. So 12 divided by negative 3 is negative 4. And we're going to evaluate that from at 2 and at 1. So this is going to be equal to. When we evaluated it at 2 at the upper bound of our interval, it's gonna be negative 4 times 2 to the negative 3 power. So it's negative 4 times what is that 2, that's 1 over 2 to the 3rd over times one-eighth is one way to think about that. And then we're going to have minus this evaluated one. So minus negative 4 times t to the negative 3. Well t to the negative 3 is just 1 so it's going to be negative 4 times 1. And this is going to be equal to, we're really in the home stretch now, this is equal to, this part right over here is, negative one-half. So this is negative one-half. And this part right over here is positive four. So, positive four, minus one-half, we could wri, either write that as three and a half. Or if we wanted to write it as an improper fraction, we could write this as seven halves. So the average value of our acceleration over this interval is seven halves. If the position was given in meters and time was in seconds. And this would be seven halves meters per second squared is the average exceleration between time and one second and time and two seconds.