Answer:
![d = \dfrac{9t^{2} }{2} - \dfrac{2}{9} t + \dfrac{t^3}{3}](https://tex.z-dn.net/?f=d%20%3D%20%5Cdfrac%7B9t%5E%7B2%7D%20%7D%7B2%7D%20-%20%5Cdfrac%7B2%7D%7B9%7D%20t%20%2B%20%5Cdfrac%7Bt%5E3%7D%7B3%7D)
Step-by-step explanation:
Given the equation of velocity w.r.to time 't':
![v=9t-\dfrac{2}{9}+t^2 ...... (1)](https://tex.z-dn.net/?f=v%3D9t-%5Cdfrac%7B2%7D%7B9%7D%2Bt%5E2%20......%20%281%29)
<em>Formula for Displacement</em>:
![Displacement = \text{velocity} \times \text{time}](https://tex.z-dn.net/?f=Displacement%20%3D%20%5Ctext%7Bvelocity%7D%20%20%5Ctimes%20%5Ctext%7Btime%7D)
So, if we find integral of velocity w.r.to time, we will get displacement.
![\Rightarrow \text{Displacement}=\int {v} \, dt](https://tex.z-dn.net/?f=%5CRightarrow%20%5Ctext%7BDisplacement%7D%3D%5Cint%20%7Bv%7D%20%5C%2C%20dt)
![\Rightarrow \int {v} \, dt = \int ({9t-\dfrac{2}{9}+t^2}) \, dt \\\Rightarrow \int{9t} \, dt - \int{\dfrac{2}{9}} \, dt + \int{t^2} \, dt\\\Rightarrow s=\dfrac{9t^{2} }{2} - \dfrac{2}{9} t + \dfrac{t^3}{3} + C ....... (1)](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cint%20%7Bv%7D%20%5C%2C%20dt%20%3D%20%5Cint%20%28%7B9t-%5Cdfrac%7B2%7D%7B9%7D%2Bt%5E2%7D%29%20%5C%2C%20dt%20%5C%5C%5CRightarrow%20%5Cint%7B9t%7D%20%5C%2C%20dt%20-%20%5Cint%7B%5Cdfrac%7B2%7D%7B9%7D%7D%20%5C%2C%20dt%20%2B%20%5Cint%7Bt%5E2%7D%20%5C%2C%20dt%5C%5C%5CRightarrow%20s%3D%5Cdfrac%7B9t%5E%7B2%7D%20%7D%7B2%7D%20-%20%5Cdfrac%7B2%7D%7B9%7D%20t%20%2B%20%5Cdfrac%7Bt%5E3%7D%7B3%7D%20%2B%20C%20.......%20%281%29)
Here, C is constant (because it is indefinite integral)
<u>Formula for integration used:</u>
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Now, it is given that s = 0, when t = 0.
Putting the values in equation (1):
![0=\dfrac{9\times 0^{2} }{2} - \dfrac{2}{9}\times 0 + \dfrac{0^3}{3} + C\\\Rightarrow C = 0](https://tex.z-dn.net/?f=0%3D%5Cdfrac%7B9%5Ctimes%200%5E%7B2%7D%20%7D%7B2%7D%20-%20%5Cdfrac%7B2%7D%7B9%7D%5Ctimes%200%20%2B%20%5Cdfrac%7B0%5E3%7D%7B3%7D%20%2B%20C%5C%5C%5CRightarrow%20C%20%3D%200)
So, the equation for displacement becomes:
![s=\dfrac{9t^{2} }{2} - \dfrac{2}{9} t + \dfrac{t^3}{3}](https://tex.z-dn.net/?f=s%3D%5Cdfrac%7B9t%5E%7B2%7D%20%7D%7B2%7D%20-%20%5Cdfrac%7B2%7D%7B9%7D%20t%20%2B%20%5Cdfrac%7Bt%5E3%7D%7B3%7D)