Question

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

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

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

Your tasks is

a) Derive an equation x (t) for the instantaneous position of the particle as

a function of time using indefinite integration.

b) Sketch a graph of position vs. time over the time interval 0 ≤ t ≤ 3

seconds for Eq.1, where C=12.

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.

Answer 1

$v(t) = 3\cos(\pi t) − 2\sin(\pi t)$  

$a)$

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

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

$b)$

$c)$

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

$d)$

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

$e)$

$h = 0.5$

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

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

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

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

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