Answer:
a)
,
, b)
, c)
, d)
, e) ![v(2.602\,s) = -26.254\,\frac{m}{s}](https://tex.z-dn.net/?f=v%282.602%5C%2Cs%29%20%3D%20-26.254%5C%2C%5Cfrac%7Bm%7D%7Bs%7D)
Explanation:
a) The velocity function is determined by deriving the position function in time:
![v(t) = 25.5-9.8\cdot t](https://tex.z-dn.net/?f=v%28t%29%20%3D%2025.5-9.8%5Ccdot%20t)
Velocities after 2 seconds and 4 seconds are, respectively:
![v(2\,s) = 5.9\,\frac{m}{s}](https://tex.z-dn.net/?f=v%282%5C%2Cs%29%20%3D%205.9%5C%2C%5Cfrac%7Bm%7D%7Bs%7D)
![v(4\,s) = -13.7\,\frac{m}{s}](https://tex.z-dn.net/?f=v%284%5C%2Cs%29%20%3D%20-13.7%5C%2C%5Cfrac%7Bm%7D%7Bs%7D)
b) The maximum height is reached when velocity is equal to zero:
![25.5-9.8\cdot t = 0](https://tex.z-dn.net/?f=25.5-9.8%5Ccdot%20t%20%3D%200)
The time when the projectile reaches the maximum height:
![t = 2.602\,s](https://tex.z-dn.net/?f=t%20%3D%202.602%5C%2Cs)
c) The maximum height is:
![h (2.602\,s) = 2 + 25.5\cdot (2.602\,s)-4.9\cdot (2.602\,s)^{2}](https://tex.z-dn.net/?f=h%20%282.602%5C%2Cs%29%20%3D%202%20%2B%2025.5%5Ccdot%20%282.602%5C%2Cs%29-4.9%5Ccdot%20%282.602%5C%2Cs%29%5E%7B2%7D)
![h(2.602\,s) = 35.176\,m](https://tex.z-dn.net/?f=h%282.602%5C%2Cs%29%20%3D%2035.176%5C%2Cm)
d) The projectile hits the ground when height is equal to zero:
![-4.9\cdot t^{2}+25.5\cdot t + 2 =0](https://tex.z-dn.net/?f=-4.9%5Ccdot%20t%5E%7B2%7D%2B25.5%5Ccdot%20t%20%2B%202%20%3D0)
The roots of the second order polynomial are presented below:
![t_{1} \approx 5.281\,s](https://tex.z-dn.net/?f=t_%7B1%7D%20%5Capprox%205.281%5C%2Cs)
![t_{2} \approx -0.077\,s](https://tex.z-dn.net/?f=t_%7B2%7D%20%5Capprox%20-0.077%5C%2Cs)
The first one is the only reasonable solution in physical terms.
![t = 5.281\,s](https://tex.z-dn.net/?f=t%20%3D%205.281%5C%2Cs)
e) The velocity of the projectile when it hits the ground is:
![v(2.602\,s) = 25.5-9.8\cdot (5.281\,s)](https://tex.z-dn.net/?f=v%282.602%5C%2Cs%29%20%3D%2025.5-9.8%5Ccdot%20%285.281%5C%2Cs%29)
![v(2.602\,s) = -26.254\,\frac{m}{s}](https://tex.z-dn.net/?f=v%282.602%5C%2Cs%29%20%3D%20-26.254%5C%2C%5Cfrac%7Bm%7D%7Bs%7D)