a ball is thrown into the air by a baby alien on a planet in the system of alpha centauri with a velocity of…

a ball is thrown into the air by a baby alien on a planet in the system of alpha centauri with a velocity of 43 ft/s. its height in feet after t seconds is given by y = 43t - 20t². a. find the average velocity for the time period beginning when t=3 and lasting .01 s: .005 s: .002 s: .001 s: note: for the above answers, you may have to enter 6 or 7 significant digits if you are using a calculator. estimate the instantaneous velocity when t=3. question help: post to forum submit question

a ball is thrown into the air by a baby alien on a planet in the system of alpha centauri with a velocity of 43 ft/s. its height in feet after t seconds is given by y = 43t - 20t². a. find the average velocity for the time period beginning when t=3 and lasting .01 s: .005 s: .002 s: .001 s: note: for the above answers, you may have to enter 6 or 7 significant digits if you are using a calculator. estimate the instantaneous velocity when t=3. question help: post to forum submit question

Answer

Answer:

  • For $\Delta t = 0.01$ s: $-77.02$ ft/s
  • For $\Delta t = 0.005$ s: $-76.97$ ft/s
  • For $\Delta t = 0.002$ s: $-76.94$ ft/s
  • For $\Delta t = 0.001$ s: $-76.92$ ft/s
  • Estimated instantaneous velocity at $t = 3$: $-76.9$ ft/s

Explanation:

Step1: Recall average - velocity formula

The average velocity $v_{avg}=\frac{\Delta y}{\Delta t}=\frac{y(t + \Delta t)-y(t)}{\Delta t}$, where $y(t)=43t - 20t^{2}$.

Step2: Calculate $y(t)$ at $t = 3$

$y(3)=43\times3-20\times3^{2}=129 - 180=-51$.

Step3: Calculate $y(t+\Delta t)$ for $\Delta t = 0.01$

$y(3 + 0.01)=43\times(3 + 0.01)-20\times(3 + 0.01)^{2}=43\times3.01-20\times9.0601=129.43-181.202=-51.772$. $v_{avg}=\frac{y(3.01)-y(3)}{0.01}=\frac{-51.772+51}{0.01}=-77.02$ ft/s.

Step4: Calculate $y(t+\Delta t)$ for $\Delta t = 0.005$

$y(3 + 0.005)=43\times(3 + 0.005)-20\times(3 + 0.005)^{2}=43\times3.005-20\times9.030025=129.215-180.6005=-51.3855$. $v_{avg}=\frac{y(3.005)-y(3)}{0.005}=\frac{-51.3855 + 51}{0.005}=-76.97$ ft/s.

Step5: Calculate $y(t+\Delta t)$ for $\Delta t = 0.002$

$y(3+0.002)=43\times(3 + 0.002)-20\times(3 + 0.002)^{2}=43\times3.002-20\times9.012004=129.086-180.24008=-51.15408$. $v_{avg}=\frac{y(3.002)-y(3)}{0.002}=\frac{-51.15408 + 51}{0.002}=-76.94$ ft/s.

Step6: Calculate $y(t+\Delta t)$ for $\Delta t = 0.001$

$y(3 + 0.001)=43\times(3+0.001)-20\times(3 + 0.001)^{2}=43\times3.001-20\times9.006001=129.043-180.12002=-51.07702$. $v_{avg}=\frac{y(3.001)-y(3)}{0.001}=\frac{-51.07702+51}{0.001}=-76.92$ ft/s.

Step7: Estimate instantaneous velocity

As $\Delta t$ gets smaller, the average - velocity approaches the instantaneous velocity. So, the estimated instantaneous velocity at $t = 3$ is approximately $-76.9$ ft/s.