Math

QuestionAn object is launched with an initial velocity of 48 ft/s from 80 ft. Find average velocity from t=0t=0 to t=2t=2 and t=2t=2 to t=4t=4. Use h(t)=16t2+48t+80h(t)=-16 t^{2}+48 t+80.

Studdy Solution

STEP 1

Assumptions1. The object is launched upward with an initial velocity of48 feet per second. . The object is launched from a height of80 feet.
3. The height of the object tt seconds after it is launched is given by the function h(t)=16t+48t+80h(t)=-16 t^{}+48 t+80.

STEP 2

First, we need to find the average velocity in the first two seconds after the object is launched. The average velocity is given by the change in position (or height in this case) divided by the change in time. This can be written asAveragevelocity=h(2)h(0)20Average\, velocity = \frac{h(2) - h(0)}{2 -0}

STEP 3

Now, plug in the given function h(t)h(t) to calculate h(2)h(2) and h(0)h(0).
h(2)=16(2)2+48(2)+80h(2) = -16(2)^{2} +48(2) +80h(0)=16(0)2+48(0)+80h(0) = -16(0)^{2} +48(0) +80

STEP 4

Calculate the values of h(2)h(2) and h(0)h(0).
h(2)=16(4)+96+80=112h(2) = -16(4) +96 +80 =112h(0)=80h(0) =80

STEP 5

Now that we have the values of h(2)h(2) and h(0)h(0), we can substitute these into the average velocity formula.
Averagevelocity=112802Average\, velocity = \frac{112 -80}{2}

STEP 6

Calculate the average velocity in the first two seconds after the object is launched.
Averagevelocity=322=16feet/secondAverage\, velocity = \frac{32}{2} =16\, feet/second

STEP 7

Next, we need to find the average velocity from t=2t=2 to t=4t=4 seconds. The average velocity is given by the change in position (or height in this case) divided by the change in time. This can be written asAveragevelocity=h(4)h(2)42Average\, velocity = \frac{h(4) - h(2)}{4 -2}

STEP 8

Now, plug in the given function h(t)h(t) to calculate h(4)h(4) and h(2)h(2).
h(4)=16(4)2+48(4)+80h(4) = -16(4)^{2} +48(4) +80h(2)=112h(2) =112 (from previous calculation)

STEP 9

Calculate the value of h(4)h(4).
h(4)=16(16)+192+80=h(4) = -16(16) +192 +80 =

STEP 10

Now that we have the values of h(4)h(4) and h(2)h(2), we can substitute these into the average velocity formula.
Averagevelocity=01122Average\, velocity = \frac{0 -112}{2}

STEP 11

Calculate the average velocity from t=t= to t=4t=4 seconds.
Averagevelocity=112=56feet/secondAverage\, velocity = \frac{-112}{} = -56\, feet/secondThe average velocity in the first two seconds after the object is launched is16 feet per second, and the average velocity from t=t= to t=4t=4 seconds is -56 feet per second.

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