For the following position functions, make a table of average ...

Question

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time.

c. s(t)=40 sin 2t at t=0

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with velocities. This problem says, considering the position function S of T is equal to 80 multiplied by the sine of the quantity of three T. We need to approximate the instantaneous velocity at T equal to zero by finding the average velocities at T equal to 10.50 0.10 0.01 and 0.001. We also need to provide a table for the average velocities and time intervals. So the first thing we're going to do is actually create this table. So we're gonna have a table that has two rows, one, it's going to the first row is going to have our time intervals in it. And our second row is going to contain our average velocities. So for our time intervals, we are actually given that in the problem. And since we're trying to find the instantaneous velocity at T equal to zero, all of our time intervals are gonna start at zero. And our upper limit for that interval is actually going to change. So our first time interval is going to be going from 0 to 1. Our next time interval is going to be going from zero to 0.5. The next time interval is gonna be going from zero to 0.1. The next time interval it's going to be going from zero to 0.01. And the final time interval is going to be going from zero to 0.001. And so now all we need to do is calculate the average velocities for each of those time intervals to fill in this table. So we're gonna start off with the first time interval which is going to be going from 0 to 1. So here we just need to calculate the average velocity over this time interval. And so we call your definition for the average velocity over a time interval. That's just gonna be our change of position divided by the change in time. So we'll to note our average velocity as the average and this is going to be equal to our final position, which is gonna be our position when T is equal to one. So it's gonna be s of one minus our initial position, which is going to be um our position when T is equal to zero. So it's gonna be s of zero and that quantity is gonna be divided by our change in time, which is just gonna be one minus zero. Now, since all of our time intervals start at T equal to zero, it will save us some time if we go ahead and calculate the value for s of zero, since it's going to appear up in every average velocity calculation for every interval. So we'll do that now. So we'll have s of zero, that's gonna be equal to. And here we'll use the equation that we were given in the problem that's gonna be equal to 80 multiplied by the sine of the quantity of three multiplied by zero. And so when I evaluate this um expression, this is going to be equal to zero is three multiplied by 00. And the sign of zero is zero. So that means sub zero is always gonna be zero. So the only thing that we really need to calculate is the position for when T is equal to the upper limit of our time interval. So doing that, we have the average is equal to, for s of one, we're going to have 80 multiplied by the sine of the quantity of three multiplied by one in quantity minus for S of zero, we're gonna have zero and that's gonna be divided by one by zero, which is just one. And so we can evaluate this expression. And when we do so this is going to be equal to 11.290. And here I've just rounded my answer to three decimal places. So that is going to be our average velocity for this first time interval. And so we'll add that value into the first column of our table. So we'll write 11.290. So now all we need to do is repeat this calculation for the other four time intervals. So the next interval that we're going to look at is going to be the interval going from zero to 0.5. And here we're going to calculate the average velocity the same way. So we'll have the average is equal to s of 0.5 minus s of zero. That quantity is divided by the quantity of 0.5 minus zero. So again, substituting our expression or our position for the s of 0.5. So we'll have the average is equal to 80 multiplied by the sine of the quantity of three multiplied by 0.5 in quantity minus zero. That quantity is divided by 0.5 minus zero, which is 0.5. To evaluate this expression, we will get 159.5 99. Here again, we've just rounded to three decimal places. So we'll add that into the next column of our table. So we'll have 159.599 written in that column. And so that is gonna be the average velocity for that time interval. We'll now move on to the next time interval, which is going to be going from zero to 0.1. And again, we're going to be repeating the same calculation that we have the average that's gonna be equal to s of 0.1 minus s of zero. That quantity is gonna be divided by the quantity of 0.1 minus zero. Again, we'll plug in our expression for our position for s of 0.1. So have the average, it's equal to 80 multiplied by the sign of the quantity of three multiplied by 0.1 in quantity minus zero. That quantity is divided by 0.1 minus zero, which is just 0.1. And so again, we can evaluate this expression to calculate our average velocity and we will get 236.416. Again rounding to just three decimal places. So that is our average velocity for this time interval. And so we will add it to the next column in our table that is 236.416. So now we need to look at the next time interval which is going to be going from zero 20.01. Let me set RV average gonna be equal to s of 0.01 minus s of zero. That quantity is gonna be divided by the quantity of 0.01 minus zero. So again, plugging in for expression, for a position will have the average is equal to 80 multiplied by the sign of the quantity of three multiplied by 0.01 in quantity minus zero, that quantity is divided by 0.01 since we have minus zero, which was just giving you back 0.01. So evaluating this expression, we will get 239.964 again rounded to just three decimal places. So we'll add that value into our table. So we'll add in 239.964 into the next column and the final interval that we need to calculate, it's going to be going from zero 20.001. So here again, we'll just calculate the average velocity like we've done for the other intervals. So we'll have the average is equal to s of 0.001 minus s of zero. And that quantity is gonna be divided by the quantity of 0.001 minus zero. Again, substituting in our formula for the possession, we have the average is equal to 80 multiplied by the sign of the quantity of three multiplied by 0.001 in quality minus zero. And that quantity is divided by 0.001 minus zero, which is 0.001. Again, when we evaluate this expression, we will get 240.000. So that is the average velocity for this time val. And here again, we've just rounded to three decimal places. We will come back up to our table and add that into the final column. So in the final column, we'll have 240 0.000. So now that we have all of our values into our table, we can use it to calculate our instantaneous velocity. So we'll label that as V instantaneous and our instantaneous velocity is going to be um the velocity when our time interval is approaching zero. So here for average velocities as we decrease the size of our time interval, it looks like it's approaching a certain value. And in this case, that value is gonna be 240. So that means our instantaneous velocity, V instantaneous is going to be approximately equal to 240 and that occurs at t equal to zero. So that's gonna be our answer for the instantaneous velocity. So here just to recap of what we've done, we first off um started by creating the table that contained our time intervals. And in order to fill out this table, we need to calculate the average velocity over each of those time intervals. So we did that using our definition for average velocity and we repeated that calculation five times one for each of the time intervals. Then once we had all of our average velocities calculated, we looked at the trend of those average velocities as our time intervals began to shrink as we got smaller and smaller, we began to approach a particular value. And for this case, our instantaneous velocity, which is the value that the average velocity is approaching when their time manifold goes to zero turned out to be 240. So I hope that this explanation has been useful and I'll see you in the next video.

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time.

c. s(t)=40 sin 2t at t=0

Key Concepts

Average Velocity

Instantaneous Velocity

Trigonometric Functions in Motion

Watch next