Answer:
The question is incomplete, the complete question is "A car drives on a circular road of radius R. The distance driven by the car is given by d(t)= at^3+bt [where a and b are constants, and t in seconds will give d in meters]. In terms of a, b, and R, and when t = 3 seconds, find an expression for the magnitudes of (i) the tangential acceleration aTAN, and (ii) the radial acceleration aRAD3"
answers:
a.![18a m/s^{2}](https://tex.z-dn.net/?f=18a%20m%2Fs%5E%7B2%7D)
b. ![a_{rad}=\frac{(27a +b)^{2}}{R}](https://tex.z-dn.net/?f=a_%7Brad%7D%3D%5Cfrac%7B%2827a%20%2Bb%29%5E%7B2%7D%7D%7BR%7D)
Explanation:
First let state the mathematical expression for the tangential acceleration and the radial acceleration.
a. tangential acceleration is express as
![a_{tan}=\frac{d|v|}{dt} \\](https://tex.z-dn.net/?f=a_%7Btan%7D%3D%5Cfrac%7Bd%7Cv%7C%7D%7Bdt%7D%20%5C%5C)
since the distance is expressed as
![d=at^{3}+bt](https://tex.z-dn.net/?f=d%3Dat%5E%7B3%7D%2Bbt)
the derivative is the velocity, hence
![V(t)=\frac{dd(t)}{dt}\\V(t)=3at^{2}+b\\](https://tex.z-dn.net/?f=V%28t%29%3D%5Cfrac%7Bdd%28t%29%7D%7Bdt%7D%5C%5CV%28t%29%3D3at%5E%7B2%7D%2Bb%5C%5C)
hence when we take the drivative of the velocity we arrive at
b. the expression for the radial acceleration is expressed as
![a_{rad}=\frac{v^{2}}{r}\\a_{rad}=\frac{(3at^{2} +b)^{2}}{R}\\t=3\\a_{rad}=\frac{(27a +b)^{2}}{R}](https://tex.z-dn.net/?f=a_%7Brad%7D%3D%5Cfrac%7Bv%5E%7B2%7D%7D%7Br%7D%5C%5Ca_%7Brad%7D%3D%5Cfrac%7B%283at%5E%7B2%7D%20%2Bb%29%5E%7B2%7D%7D%7BR%7D%5C%5Ct%3D3%5C%5Ca_%7Brad%7D%3D%5Cfrac%7B%2827a%20%2Bb%29%5E%7B2%7D%7D%7BR%7D)
The required value of the index of refraction of glass is 1.5.
Given data:
The speed of light in a vacuum is,
.
The speed of light in a glass is,
.
Light has the tendency to travel from one medium to another, then the difference between the speeds of light in various mediums is determined by a term, known as the index of refraction. The mathematical expression for the index of refraction of glass is,
n = c / v
Solving as
![n = \dfrac{3.0 \times 10^{8}}{2.0 \times 10^{8}}\\\\n = 1.5](https://tex.z-dn.net/?f=n%20%3D%20%5Cdfrac%7B3.0%20%5Ctimes%2010%5E%7B8%7D%7D%7B2.0%20%5Ctimes%2010%5E%7B8%7D%7D%5C%5C%5C%5Cn%20%3D%201.5)
Thus, we can conclude that the required value of the index of refraction of glass is 1.5.
Learn more about the index of refraction here:
brainly.com/question/17156275