1) Average speed in the different intervals:
48.42 m/s
49.36 m/s
50.10 m/s
50.0 m/s
50.0 m/s
2) Instantaneous speed at t = 1 s is 50.28 m/s
Step-by-step explanation:
1)
The position of the arrow at time t is given by the equation
![y(t)=54t-1.86t^2](https://tex.z-dn.net/?f=y%28t%29%3D54t-1.86t%5E2)
The average speed can be found by dividing the distance covered by the time taken:
![v=\frac{\Delta y}{\Delta t}](https://tex.z-dn.net/?f=v%3D%5Cfrac%7B%5CDelta%20y%7D%7B%5CDelta%20t%7D)
(i) For this interval [1,2]:
![y(1)=54(1)-1.86(1)^2=52.14 m\\y(2)=54(2)-1.86(2)^2=100.56 m](https://tex.z-dn.net/?f=y%281%29%3D54%281%29-1.86%281%29%5E2%3D52.14%20m%5C%5Cy%282%29%3D54%282%29-1.86%282%29%5E2%3D100.56%20m)
So ![\Delta y = y(2)-y(1)=100.56-52.14=48.42 m](https://tex.z-dn.net/?f=%5CDelta%20y%20%3D%20y%282%29-y%281%29%3D100.56-52.14%3D48.42%20m)
Time interval is ![\Delta t = 2-1 = 1 s](https://tex.z-dn.net/?f=%5CDelta%20t%20%3D%202-1%20%3D%201%20s)
So the average speed is
![v=\frac{48.42}{1}=48.42 m/s](https://tex.z-dn.net/?f=v%3D%5Cfrac%7B48.42%7D%7B1%7D%3D48.42%20m%2Fs)
(II) For this interval [1,1.5]:
![y(1)=54(1)-1.86(1)^2=52.14 m\\y(1.5)=54(1.5)-1.86(1.5)^2=76.82 m](https://tex.z-dn.net/?f=y%281%29%3D54%281%29-1.86%281%29%5E2%3D52.14%20m%5C%5Cy%281.5%29%3D54%281.5%29-1.86%281.5%29%5E2%3D76.82%20m)
So ![\Delta y = y(1.5)-y(1)=76.82-52.14=24.68 m](https://tex.z-dn.net/?f=%5CDelta%20y%20%3D%20y%281.5%29-y%281%29%3D76.82-52.14%3D24.68%20m)
Time interval is ![\Delta t = 1.5-1 = 0.5 s](https://tex.z-dn.net/?f=%5CDelta%20t%20%3D%201.5-1%20%3D%200.5%20s)
So the average speed is
![v=\frac{24.68}{0.5}=49.36 m/s](https://tex.z-dn.net/?f=v%3D%5Cfrac%7B24.68%7D%7B0.5%7D%3D49.36%20m%2Fs)
(iii) For this interval [1,1.1]:
![y(1)=54(1)-1.86(1)^2=52.14 m\\y(1.5)=54(1.1)-1.86(1.1)^2=57.15 m](https://tex.z-dn.net/?f=y%281%29%3D54%281%29-1.86%281%29%5E2%3D52.14%20m%5C%5Cy%281.5%29%3D54%281.1%29-1.86%281.1%29%5E2%3D57.15%20m)
So ![\Delta y = y(1.1)-y(1)=57.15-52.14=5.01 m](https://tex.z-dn.net/?f=%5CDelta%20y%20%3D%20y%281.1%29-y%281%29%3D57.15-52.14%3D5.01%20m)
Time interval is ![\Delta t = 1.1-1 = 0.1 s](https://tex.z-dn.net/?f=%5CDelta%20t%20%3D%201.1-1%20%3D%200.1%20s)
So the average speed is
![v=\frac{5.01}{0.1}=50.10 m/s](https://tex.z-dn.net/?f=v%3D%5Cfrac%7B5.01%7D%7B0.1%7D%3D50.10%20m%2Fs)
(iv) For this interval [1,1.01]:
![y(1)=54(1)-1.86(1)^2=52.14 m\\y(1.5)=54(1.01)-1.86(1.01)^2=52.64 m](https://tex.z-dn.net/?f=y%281%29%3D54%281%29-1.86%281%29%5E2%3D52.14%20m%5C%5Cy%281.5%29%3D54%281.01%29-1.86%281.01%29%5E2%3D52.64%20m)
So ![\Delta y = y(1.01)-y(1)=52.64-52.14=0.50 m](https://tex.z-dn.net/?f=%5CDelta%20y%20%3D%20y%281.01%29-y%281%29%3D52.64-52.14%3D0.50%20m)
Time interval is ![\Delta t = 1.01-1 = 0.01 s](https://tex.z-dn.net/?f=%5CDelta%20t%20%3D%201.01-1%20%3D%200.01%20s)
So the average speed is
![v=\frac{0.50}{0.01}=50.0 m/s](https://tex.z-dn.net/?f=v%3D%5Cfrac%7B0.50%7D%7B0.01%7D%3D50.0%20m%2Fs)
(v) For this interval [1,1.001]:
![y(1)=54(1)-1.86(1)^2=52.14 m\\y(1.5)=54(1.001)-1.86(1.001)^2=52.19 m](https://tex.z-dn.net/?f=y%281%29%3D54%281%29-1.86%281%29%5E2%3D52.14%20m%5C%5Cy%281.5%29%3D54%281.001%29-1.86%281.001%29%5E2%3D52.19%20m)
So ![\Delta y = y(1.001)-y(1)=52.19-52.14=0.05 m](https://tex.z-dn.net/?f=%5CDelta%20y%20%3D%20y%281.001%29-y%281%29%3D52.19-52.14%3D0.05%20m)
Time interval is ![\Delta t = 1.001-1 = 0.001 s](https://tex.z-dn.net/?f=%5CDelta%20t%20%3D%201.001-1%20%3D%200.001%20s)
So the average speed is
![v=\frac{0.05}{0.001}=50.0 m/s](https://tex.z-dn.net/?f=v%3D%5Cfrac%7B0.05%7D%7B0.001%7D%3D50.0%20m%2Fs)
2)
In this part we want to estimate the instantaneous speed of the arrow. The instantaneous speed can be found by calculating the derivative of the position, therefore:
![y(t)=54t-1.86t^2](https://tex.z-dn.net/?f=y%28t%29%3D54t-1.86t%5E2)
So the derivative is
![v(t)=y'(t)=54-2\cdot 1.86t=54-3.72t](https://tex.z-dn.net/?f=v%28t%29%3Dy%27%28t%29%3D54-2%5Ccdot%201.86t%3D54-3.72t)
And by substituting t = 1 s, we find the instantaneou speed at that time:
![v(1)=54-3.72(1)=50.28 m/s](https://tex.z-dn.net/?f=v%281%29%3D54-3.72%281%29%3D50.28%20m%2Fs)
Learn more about speed:
brainly.com/question/8893949
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