The position of an object moving vertically along a line is gi...

Question

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-4.9t^2+30t+20$ . Find the average velocity of the object over the following intervals.

$\left\lbrack0,h\right\rbrack$ , where $h\gt{0}$ is a real number

Accepted Answer

All right, everyone. Let's take a look at this practice problem dealing with average velocity. This problem says a ball is thrown vertically with its position over time given by S of T is equal to minus 5.9 T squared plus 36 T plus 10. What is the average velocity of the ball over the interval going from zero to k where K is greater than zero is a real number. We give four possible choices as our answers. For choice A we have minus 11.8 K minus 18. For choice B, we have minus 11.8 K plus 18. For choice C, we have minus 5.9 K plus 36. And for choice D, we have minus 5.9 K minus 36. So here we need to calculate the average velocity of the ball over this inter interval going from zero to K for this problem. So we're going to use our definition of average velocity. So recall that our average velocity is equal to our change in position divided by our change in time. So we'll have our average velocity which will represent as the average, that's gonna be equal to our change in our position. So that's gonna be our position at time K. So that'll be s of K minus our initial position, which is going to be at T equal to zero. So it'll be s of zero and that quantity is gonna be divided by our change in time, which is just gonna be K minus zero. Now, what we can do is click in our expression or s of T into this equation for each of our times. So we'll have the average equal to or S of K that's gonna be minus 5.9 K squared plus 36 K plus 10. And then we subtract from that s of zero. So I'll have minus the quantity or S zero, I'll have minus 5.9 multiplied by zero squared plus 36 multiplied by zero plus 10. And that whole expression is going to be divided by K minus zero, which is just K. So we can simplify our numerator. So the minus 5.9 multiplied by zero squared. That term is zero since anything multiplied by zero is zero, likewise, the 36 multiplied by zero is also zero. I'll then also have a plus 10 minus 10. So those two values of 10 cancel out and then I can divide the um other expressions by K in our denominator. So simplifying that will have the average is equal to minus 5.9. And then here I have K squared divided by K which just gives me K, they don't have plus 36 K divided by K, which is just 36. So this is the answer that I am actually looking for. And when I compare that to my answer choices, the correct answer choice here is c so I hope this explanation has been useful and I'll see you in the next video.

The position of an object moving vertically along a line is given by the function $s\(\left\)(t\(\right\))=-4.9t^2+30t+20$ . Find the average velocity of the object over the following intervals.
$\left\[\lbrack\)0,h\(\right\]\rbrack$ , where $h\(\gt{0}\)$ is a real number

Key Concepts

Position Function

Average Velocity

Intervals and Limits

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