Question

Given the following table of velocity data 1, 0 05 10 15 20 25 3.0 3.5 4.0 V m/s 0 12 16 1.4 2.0 2.0 1.8 1.6 1.3 a) Estimate the position of the vehicle at 1.5 seconds, as accurately as possible b) Estimate the acceleration of the vehicle at 1.5 seconds, as accurately as c) Estimate the position of the vehicle at 4 seconds, as accurately as possible d) Estimate the acceleration of the vehicle at 4 seconds, as accurately as possible Show your work & describe your steps in the space provided on this page. Please work neatly to ensure full credit. Write your answers in the boxes provided, with units. (5 pts cach)

Answer 1

x	f(x)
0	0
0.5	1.2
1.0	1.6
1.5	1.4
2.0	2.0
2.5	2.0
3.0	1.8
3.5	1.6
4.0	1.3

(1) Using Newton's forward/backward differentiation method to find solution

[dV/dt] at (t = t₀ )=1/h⋅(ΔY₀-1/2⋅Δ²Y₀+1/3⋅Δ³Y₀-1/4⋅Δ⁴Y₀)

put h = 0.5

and value of  ΔY_0, Δ²Y₀ , Δ³Y₀ , Δ⁴Y₀  accordingly.

We will get

dV/dt = 1.333333 at t=1.5

${\color{Red} }$ (4)

for acceleration at t = 4

using Newton's backward differentiation table is as follows.

[dVd/t] (t=t_n)=1/h⋅(∇Y_n+ 1/2⋅∇^₂Y^_n+1/3⋅∇³Y_n+1/4⋅∇⁴Y_n)

pt h = 0.5 and value of

∇Y_n, ∇^₂Y_n∇³Y_n, ∇⁴Y_n accordingly

we will get

dVd/t = -0.91667 at t = 4

(2) using newtons simpson 1/3 rule rule

location at t = 4 is X = 6.15

(3)  using newtons simpson 1/3 rule rule

  location at t = 1.5 is X = 1.56667

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