Average velocity The velocity in m/s of an ob...

Question

Average velocity The velocity in m/s of an object moving along a line over the time interval [0,6] is v (t) = t² + 3t. Find the average velocity of the object over this time interval.

Accepted Answer

Welcome back, everyone. An object's velocity in meters per second is described by V of T equals 4 C minus T2 fort in the interval from 2 to 7 inclusive. Calculate the average velocity over this interval. So for this problem, let's recall the average value formula. F average is equal to 11 divided by B minus A. Where A and B are represented by the interval from A to B inclusive, multiplied by the integral from A to B of F of XD X. So we can apply this concept to our problem. We can basically say that the average velocity, the average is going to be equal to 1 divided by B minus A. B is the upper limit that be 7. A is the lower limit that would be 2, and we integrate from 2 to 7. And we're integrating V of TDT, right? So that'd be 4 T minus T 2 DT. So we get 1 divided by 5. And now we are going to integrate, right? So the integral of 4 T is going to be 4 multiplied by T2 divided by 2 based on the power rule, and we're subtracting T cubed divided by 3. Once again, we're applying the power rule. We're adding 1 to the original exponent and dividing by the new exponent. And now we want to evaluate the result from 2 to 7. So we get 15th, in print Cs we can simplify, specifically, we get 2 T squad minus T cubed. Divided by 3. And now let's evaluate this result from 2 to 7. To make it simpler, we can also factor out, right? And what we can do is simply factor out T2, so that'll be T2 divided by 5, and C 2 minus T divided by 3. We're evaluating from 2 to 7 and then we are ready to substitute 7, so that'd be 7 squared. Divided by 5 in 2 minus. 7 divided by 3. And we're subtracting the value of the function at 2, so that'd be 2 squared. Divided by 5, multiplied by 2 minus 2 divided by 3. Now let's simplify. We get 49 divided by 5. Which is multiplied by. 2 minus 7/3, so that would be 6/3 minus 7/3, and this is equal to -1/3, right? And we're subtracting 4/5. Multiplied by 2 minus 2/3. So 6/3 minus 2/3 is 4/3. And now let's simplify. So now for the first product we get minus 49 divided by 15. And we are subtracting 16 divided by 15. This is equal to -65 divided by 15. And now let's go ahead and divide both sides of this fraction by 5, so we can simplify, and this is going to be equal to -13 divided by 3. Our units are meters per second, right? So we can state that our final answer is -13 divided by 3 m per second. Thank you for watching.

Average velocity The velocity in m/s of an object moving along a line over the time interval [0,6] is v (t) = t² + 3t. Find the average velocity of the object over this time interval.

Key Concepts

Average Velocity

Velocity Function

Integration

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