Consider the position function s(t)=−16t^2+100t. Complete the ...

Question

Consider the position function s(t)=−16t^2+100t. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=3.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with velocities. This problem says, consider the position function S of T is equal to minus 20 T squared plus 120 T approximate the attains velocity at T equal to two by completing the given table with the average velocities. The question, we're given a table that contains two rows. First row is labeled time interval and contains the time intervals of 1 to 2, 1.5 to 21.9 to 21.99 to 2 and 1.999 to 2. The next row is labeled as average velocity and the rest of the columns are left blank. We're also given four possible choices as our answers. For choice. A we have the average velocities of 60 50 42 40.2 and 40.02. The ascertains velocity is 40 choice B. We have the average velocity of 60 55 52 50.2 and 50.02. The instantaneous velocity is 50 for choice C, the average velocities are 80 70 62 60.2 60.02. And the instantaneous velocity is 60. For choice D, the average velocities are 60 50 42 40.2 40.02 the instantaneous velocity is 160. So in order to calculate the instantaneous velocity at T equal to two, we need to first calculate the average velocities for the intervals that are given in the table. Now, for these time intervals, we're gonna start off with the first one, which is going from 1 to 2. So we need to calculate the average velocity over this time interval. And so recall your formula for the average velocity is just gonna be our change in position divided by our change in time over the time interval. So we'll la uh denote our average velocity as the average and it is going to be equal to our change of position, which is going to be our position at equal to two, which is S of two minus our position at T equal to one, which is S of one. And that quantity is gonna be divided by our change in time, which is just gonna be two minus one. So in order to evaluate this expression, I will need to find our positions at T equal to one and T equal to two. So we're going to start off with T equal to one. So we'll have s of one and that's gonna be equal to using the formula that we're given in the problem minus 20 multiplied by one squared plus 120 multiplied by one. And this is going to be equal to when we evaluate that expression to 100. So we're going to do something similar, but except this time, we're gonna have t equal to two. So we'll have s of two and that's gonna be equal to minus 20 multiplied by two squared plus 120 multiplied by two. And this is going to be equal to 160. So we can substitute those values back into our formula for the average velocity. Do we have the average, that's gonna be equal to 160 or s of two minus or s of one will have 100. And that quantity is going to be divided by two minus one, which is just one. So evaluating this expression will get our average velocity for this time an to be equal to 60. And so that is the value that I'm looking for for this interval. And we'll actually copy that into our table in the first empty column. So we're going to repeat this process or the other four time intervals. So our next time interval is going to be going from 1.5 to two. And again, we're going to be calculating the average velocity using the same formula that we just use. So we'll have the average, this could be equal to s of two minus s of 1.5. That quantity is gonna be divided by the quantity of two minus 1.5. Now, we've already calculated the value for S of two, that's gonna be 160. So we do not need to recalculate that, but we will need to calculate S of 1.5. So doing so we'll have S of 1.5 and that's gonna be equal to minus 20 multiplied by 1.5 squared plus 120 multiplied by 1.5. And I can evaluate this expression and I will get 135. So I can substitute that value back into our average average velocity formula. So we'll have the average and this is gonna be equal to 160 minus 135. That quantity is gonna be divided by two minus 1.5 which is 0.5. Evaluating this expression. We get our average velocity is equal to 50 for this time interval. And so we'll copy that value into the next column in our table. We'll write 50 into that column. So we'll continue on to the next time interval, which is the interval going from 1.9 to 2. And again, we're going to repeat the same calculation here for our average velocity. So we'll have the average, that's gonna be equal to s of two minus s of 1.9. That quantity is gonna be divided by the quantity of two minus 1.9. We're gonna again calculate our position four T equal to 1.9. We'll have s of 1.9 and that's gonna be equal to minus 20 multiplied by 1.9 squared plus 120 multiplied by 1.9. And evaluating this expression, I'll get 155 0.8. So substituting that value back into our average velocity formula, we'll have the average could be equal to 160 minus 155.8 divided by. And here we have two minus 1.9 which is 0.1. So evaluating this expression, this is gonna be equal to 42. So that is the average value for uh our, our average velocity value for this time interval. So we'll add it to the next column in our table. So next uh time info that we're going to be looking at is the interval going from 1.99 to two. Again, we're gonna repeat same calculation. So we'll have the average equal to s of two minus s of 1.99. And that quantity is gonna be divided by the quantity of two minus 1.99. We're going to calculate our position at t equal to 1.99. So we'll have s of 1.99 that's gonna be equal to minus 20 multiplied by 1.99 squared plus 120 multiplied by 1.99. And evaluating that expression will get 100 59 0.598. So we'll substitute that value back into our fora for the average velocity. So we'll get the average is equal to 160 minus 159.598. That quantity is divided by two minus 1.99 which is gonna be 0.01. Evaluate this expression. We're going to get that, our average velocity is equal to 40.2. So that is the average velocity for this time interval. And so we'll add that onto our table. So in the next column, we'll write 40.2. And so finally, we have our last time interval, which is going to be the interval going from 1.999 to 2. And again, we're gonna do the same thing as we've done for the other calculations. So we'll have the V average is equal to s of two minus s of 1.999. That quantity is gonna be divided by the quantity of two minus 1.999. Again, we'll need to calculate our quantity s of 1.999. And this is gonna be equal to minus 20 multiplied by 1.999 squared plus 120 multiplied by 1.999 evaluating this will get 159.95998. So we'll take that value and substitute in to our form for the average velocity we'll have, the average is equal to 160 minus 159 0.95 998. That quantity is divided by two minus 1.999. And that gives me 0.001. So evaluating this expression, this is gonna be equal to 40.02. And so that's gonna be our average velocity for this time interval. So we can add that to the last column in our table. In the last column, we'll write 40.02. So now at this point, I have completed our table with the average velocity. So we now need to approximate the instantaneous velocity at T equal to two. And so for an instantaneous velocity that occurs when the time interval is approaching zero. And for each of the average velocities that we've calculated the time interval has gotten smaller and smaller. So if I look at the trend at which those average velocities are going towards, it looks like that when T is equal to two, that the instantaneous velocity is going to be equal to 40. So we're gonna set our instantaneous velocity V equal to 40 at T equal to two. And so that is gonna be the answer for the instantaneous velocity. And so now if I take the values in the table as well as our value that we calculated for the instantaneous velocity and look at our answer choices, you correct answer choice corresponds to answer choice. A so just a quick little recap of what we've done for the average velocities, we had to calculate the average velocities over each of the time intervals. So we had to repeat this calculation a total of five times. Then we had to use those average velocity calculations to approximately to approximate the instantaneous velocity. And so we looked at the trend of the average velocities as the time interval got smaller. And in this case, our instantaneous velocity actually um or those average velocities actually approached the instantaneous velocity of 40 at T equal to two. So I hope that this explanation has been useful and I'll see you in the next video.

Consider the position function s(t)=−16t^2+100t. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=3. <IMAGE>

Key Concepts

Position Function

Average Velocity

Instantaneous Velocity

Watch next