Question #180624

The acceleration of a moving object can be modelled by the following

equation.


a = 3t^2


i) Using integration v = ∫ a dt to find the value of the velocity value of

the velocity at 0 ≤ t ≤ 2.2 (s).

ii) Using trapezium rules (n=6) to find the mean value of the velocity over

a range of 0 ≤ t ≤ 2.2.

iii) Using a computer spreadsheet to increase the number of intervals to

n=12 for the numerical method and compare your obtained results

with (i) and (ii).

iv) Evaluate all obtained results (form i to iv steps) using technically correct

language and a logical structure to identify whether the size of

numerical steps (intervals) and applied various numerical techniques

have affected the obtained result.


1
Expert's answer
2021-04-15T06:50:42-0400

a=3t2a = 3t²


(i) v=a dt=3t2 dt=t3v = \int a\ dt = \int3t²\ dt = t³

v(0)=03=0m/sv(2.2)=2.23=10.648m/sv(0) = 0³ = 0m/s\\ v(2.2) = 2.2³ = 10.648m/s


(ii) 02.23t2 dt\int^{2.2}_{{{0}}}3t² \ dt

f(x)=3t2,a=0,b=2.2f(x) = 3t², \quad a={ {{0}} } , \quad b = 2.2

The width of each subinterval is

Δx=ban=2.26=0.37Δ x = \dfrac{b − a}{ n} = \dfrac{2.2}{ 6}=0.37

so the grid points have the coordinates

xi=i0.37x _ i = i 0.37

Calculate the values of the function f(x) at the points xi:

f(x0)=f(0)=3(02)=02=0{f\left( {{x_0}} \right) }={ f\left( 0 \right) }={ {3}{( {{0 ²)}} }}={ {0^2} }={ 0}

f(x1)=f(0.37)=3(0.372)=0.4107{f\left( {{x_1}} \right) }={ f\left( {0.37} \right) }={ {3}(0.37²}) = 0.4107

f(x2)=f(2(0.37))=3(0.742)=1.6428{f\left( {{x_2}} \right) }={ f\left( {2(0.37)} \right) }={ {3}(0.74²}) = 1.6428

f(x3)=f(3(0.37))=3(1.112)=3.6963{f\left( {{x_3}} \right) }={ f\left( {3(0.37)} \right) }={ {3}(1.11²}) = 3.6963

f(x4)=f(4(0.37))=3(1.482)=6.5712{f\left( {{x_4}} \right) }={ f\left( {4(0.37)} \right) }={ {3}(1.48²}) = 6.5712

f(x5)=f(6(0.37))=3(1.852)=10.2675{f\left( {{x_5}} \right) }={ f\left( {6(0.37)} \right) }={ {3}(1.85²}) = 10.2675

f(x6)=f(2(0.37))=3(2.222)=14.7852{f\left( {{x_6}} \right) }={ f\left( {2(0.37)} \right) }={ {3}(2.22²}) = 14.7852


The Trapezoidal Rule formula is written in the form;

02.23t2 dtT6=Δx2[f(x0)+2f(x1)++2f(x5)+f(x6)]=0.372[0+20.4107+21.6428+23.6963+26.5712+210.2675+214.7852]={\int^{2.2}_{{{0}}}3t²\ dt \approx {T_6} }={ \frac{{\Delta x}}{2}\left[ {f\left( {{x_0}} \right) + 2f\left( {{x_1}} \right) + \cdots }\right.}+{\left.{ 2f\left( {{x_5}} \right) + f\left( {{x_6}} \right)} \right] }={ \frac{0.37}{{2}}\left[ {0 + 2 \cdot 0.4107 + 2 \cdot 1.6428 }\right.}+{\left.{ 2 \cdot 3.6963 + 2 \cdot 6.5712 + 2 \cdot 10.2675 + 2\cdot14.7852} \right] }=

19.2986 m/s19.2986\ m/s


(iii) 02.2a dt=02.23t2 dt=[t3]02.2=2.2303=10.648 m/s\int^{2.2}_{{ {0} }} a\ dt = \int^{2.2}_{{ {0} }}3t²\ dt = [t³]^{2.2}_{ {{0} }} = 2.2³ - 0³ = 10.648 \ m/s

which coincides with the original value


(iv) all numerical steps have affected the original obtained result.


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