Question

Simpsons 1/3 rule, and Simpsons 3/8 rule, use the ollowing data to compute the work in k Pressurek Volume (0 2 336 29L4 2664 260.8 2605 2496 193.6 165.6 10 2 of 3 Ministry of Education KING ARDULATHZ Faculty of Engineering, Rabi :344 Rabigh 21911 PART III 1. The following data is provided for the velocity of an object as a function of time, ts 048 12 2024 28 32 36 (a) Using the best mmerical method available, how far does the object travel froen t0 to 28 (b) Using the best numerical method availabk, what is the objects accokration at t 28 8. 0 34.761.8 82.8 99.2 112.0 121.9 129.7 135.7 140.4 2. Chemical reactions often Sollow the model de dt where e concentration, time, reaction rate, andnreaction order, Given values of e and . k and n can be evaluated by a linear regression of the logarithm of this equation log k+nloge Use this approach along with the folkowing data to estimate& and t10 20 30 0 50 60 e 3.52 2.48 1.23 087 0.61 3 of 3

Answer 1

1. (a) To find the distance travelled, integration needs to be done from t=0 to t=28 s.

Simpsons 3/8 rule:  $\int_{x_1}^{x_4}f(x)dx = \frac{3}{8}h(y_1 + 3y_2 + 3y_3 + y_4)$

Simpsons 1/3 rule:  $\int_{x_1}^{x_3}f(x)dx = \frac{1}{3}h(y_1 + 4y_2 + y_3)$

Simpsons 3/8 rule can be applied from t=0 to t=12 s and then Simpsons 1/3 rule can be applied twice: from t=12 to t=20 and from t=20 to t=28

h = 4 s

From t=0 to t=12 s

D1- 4 (0+334.7 +3* 61.8+82.8)

D1 = 558.45 m

From t=12 to t=20 s

D2 4 (82.84 99.2 112)

D2 = 788.8 m

From t=20 to t=28

D3-ה* 4 * ) +1124 * 121.94 129.7 )

D3 = 972.4 m

Total distance travelled = D1 + D2 + D3 = 2319.65 m

(b) The acceleration can be found by numerically differentiating v with respect to t at t=28 s.

The formula used is : $y'(x) = \frac{y(x+h) - y(x-h)}{2h}$

In this case it is: $y'(t=28 s) = \frac{y(t=32s) - y(t=24s)}{8}$

Therefore acceleration =

135.7- 121.9

=1.725 m/s²

2. $\frac{dc}{dt}$ needs to be found at each point.

The formula to be used for end points: $y'(x) = \frac{-3y(x)+4y(x+h)-y(x+2h)}{2h}$ (For the final end point, h must be replaced with -h)

For other points the formula to be used is : $y'(x) = \frac{y(x+h) - y(x-h)}{2h}$

The table below is computed by doing the above calculations(substitution is not shown, assuming it is very straightforward)

t	c	dc/dt	log(-dc/dt)	log (c)
10	3.52	-0.1195	-0.9226	0.5465
20	2.48	-0.0885	-1.0531	0.3945
30	1.75	-0.0625	-1.2041	0.243
40	1.23	-0.044	-1.3565	0.0899
50	0.87	-0.031	-1.5086	-0.0605
60	0.61	-0.021	-1.6778	0.2147

Using this information the graph of the equation given in the question can be plotted. It can be done by hand or in excel. A linear graph is obtained for which both the slope and intercept can be found. The graph is shown below.

Chemical reactions 05465 3945 029008m 0.2og (c) 0.4 0.6 0.8 0.243 0.0899 0.0605 -1 1 1.4 1.6 1.8 y -0.1513x-0.7576 Series1 Linear (Series)

Upon comparing this to the equation, we see that the value of n (slope) = -0.1513

log k = -0.7576

Therefore, k = 0.1747