Question

v(t) = 3 Cos(πt) − 2Sin(πt) (Eq. 1)

2- Using a mathematical model and calculus methods (e.g. numerical and

integration methods) to solve given engineering problem (Eq. 1).

Your tasks is

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

time using an Excel sheet and trendline.

d) Using definite integration and driven equation (from c) to find the area

under curve over the time interval 0 ≤ t ≤ 3 seconds and C=12.

e) Using a mid-ordinate rule and driven equation (from c) to find the area

under curve over the time interval 0 ≤ t ≤ 3 seconds at h= 0.5.

f) 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.

.

Answer 1

Answers:-

c)

$x(t) = -0.1433t+12.385$

d)

$\int_0^3(-0.1433t+12.385)=(-0.7165t^2+12.385t)|_0^3=36.511$

$e)$

$h = 0.5$

$t_1=0.25 ; x(t_1)=11.964175;x(t_1)*h=5.9820875$

$t_2=0.75 ; x(t_2)=11.892525;x(t_2)*h=5.9462625$

$t_3=1.25 ; x(t_3)=11.820875;x(t_3)*h=5.9104375\newline t_4=1.75 ; x(t_4)=11.749225;x(t_4)*h=5.8746125\newline t_5=2.25 ; x(t_5)=11.677575;x(t_5)*h=5.8387875\newline t_6=2.75 ; x(t_6)=11.605925;x(t_6)*h=5.8029625$

$\Sigma(x(t_i)*h)=35.35515$

$\int_0^3(-0.1433t+12.385)\approx\Sigma(x(t_i)*h)=35.35515$

$f)$

$x(t) =\int{v(t)}= \int3\cos(\pi t) − 2\sin(\pi t)=$

$= \frac{3}{\pi}\sin(\pi t) + \frac{2}{\pi}\cos(\pi t)+C$