Answer to Question #190545 in Calculus for Daniel

Question

Given equation, v(t) = 3 Cos(πt) − 2Sin(πt) (Eq. 1)

● v(t) is the instantaneous velocity of the car (m/s)

● t is the time in seconds

a)Find a mathematical model (e.g. equation) to correlate position and

time using an Excel sheet and trendline.

b)Find an accurate mathematical model (e.g. equation) to correlate

position and time. To complete this task you should be able to sketch

the graph again, find the accurate equation using an excel sheet and

trendline.

c)Compare the R2 between the new and previous equations (step a) and b) ).

d) Use the driven equation x(t) from (a) and solve it using the following

numerical methods over the time interval 0 ≤ t ≤ 3 seconds at h=0.5.

I. Using the trapezium method

II. Using a Simpsons rule

Answer 1

(A)The calculus method that can be used here is by putting V=$\dfrac{dx}{dt}$ ,and than by integrating both side of the equation we can get instantaneous equation of a particle covered by the particle during the whole journey and the mathematical model that can be used is that we can put t=0 and can get instantaneous distance of particle at any time.

(B) By putting V=$\dfrac{dx}{dt}$ and integrating on both side we can get instantaneous distance of car and by $\dfrac{dv}{dt}$ we can get instantaneous acceleration of car .

Instantaneous distance Equation of car can be given as $x(t)=\dfrac{3sin({\pi}t)}{\pi}+\dfrac{2cos({\pi}t)}{\pi}$

instantaneous acceleration of car can be given as a(t)=$-3{\pi}sin({\pi}t)-2{\pi}cos({\pi}t).$

(C) The $R^2$ of the previous position is greater as compared to the new position.

(D)(i) Using trapezium method-

$x(t)=\dfrac{h}{2}(V(t)-v(0))=\dfrac{0.5}{2}(v(3)-v(0))\\[9pt] =0.25(3cos(3\pi-2sin 3\pi-3cos(0) +2sin(0)) \\[9pt] =0.25\times [3(-1)-3(1)]\\[9pt]=0.25\times -6=-1.5m$

(ii) using simpson rule-

$x(t)=\int_0^3v(t)dt$

$=\int_0^3 (3cos(\pi t)-2sin (\pi t))dt\\[9pt] =-3\pi sin\pi t-2\pi cos (\pi t)|_0^3\\[9pt]=-3\pi sin3\pi-2\pi cos3\pi-[-3\pi sin0-2\pi cos0]\\[9pt]=-2\pi (-1)+2\pi (1)\\[9pt]=4\pi$