Hello, could I get some help with a Differential Equations problem that involves Eigenvalues and Eigenvectors?The set up is:There are two toy rail cars, Car 1, and Car 2. Car 1 has a mass of 2 kg, and is traveling 3 m/s towards Car 2, which has a mass of 1 kg, and is traveling towards Car 1 at 2 m/s. There is a bumper on the second rail car that engages at the moment the cars hit (connecting Car 1 and Car 2), and does not let go. The bumper acts like a spring with spring constant K = 2 N/m. Car 2 is 7 m from the wall at the time of collision (Car 2 is between Car 1 and the wall).I have attached the work I have done so far, but I'm not understanding how to find x1(t) and x2(t), how we know Car 2 hits the wall (or moves away from it), and at what speed Car 1 travels to stay in place after link-up (given: 1 m/s, but not sure why that is).Thank you in advance. 3 m/sm, x₁c)2 X₁" = -2 X₁ + 2X₂22X₁"=-1X₁ + X₂AXDE11 = K (x₂-x₁)det (A-1)=0==ra, +(A-21) √=0² ±2=0= (-11-(7) 19HilsX (0) = a₁ - A₂=7A = -1a₁ + 6₁ (t) + 2a₂Aug. Matrix2₂=-3(a₁ + 2a₂)wall ?I m/s ?[-+2²-2-7]2-2XHE[1] (a₁ +h}) + [=1] [9₂ cos(√is t) + b₂ Sin (√5 +)22 m/sM₂ X₂= -K (X₂ X₁)1 X₂ "1= 2X₁ -2x₂TTTX₂"= 2x₁ - 2×₂T2-2= [0]=0MNH²,ㅋ[MO21→ (-1->)/(2-7) -(1) (2) =2₁0 2₁₂ = -3W₁ = 0, W₂ = √3-a₂ cos (√3+)-by sin (√5 +)]сос(3+)Sin (√3+)+262⇒ V₁ = √₂, J₁= [1][ 27 V₁ =- = √₂2, 2²/= [ 21 ]7 MALAM,=>010]displacementsprend7 = [b₁ (f) - b₂ sin (√3+) = x ' (f) = [ 6₁ - 53 62₂ 405 (18+)26₁ +213 12₂ c05 (√3+)(b₁ (f) +26₂ Sin√(√3+))Алд, тархаRREF2²(₂). [1-41 ] - [ ] = [101]==3253a₁ =092=0=>a) x² = [ { + @ (+1)(-25) 51 (28)] = [ { + + + + 51 (²+)}]}]Sin (√3+))353 Sin (√3+)3+ +) (2) (-3/3) Sin (√3+)]을 +3-√3 Sin (√3+)wall6, < 1/b₂ = -13√3

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Answered: Hello, could I get some help with a…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

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Hello, could I get some help with a Differential Equations problem that involves Eigenvalues and Eigenvectors?

The set up is:

There are two toy rail cars, Car 1, and Car 2. Car 1 has a mass of 2 kg, and is traveling 3 m/s towards Car 2, which has a mass of 1 kg, and is traveling towards Car 1 at 2 m/s. There is a bumper on the second rail car that engages at the moment the cars hit (connecting Car 1 and Car 2), and does not let go. The bumper acts like a spring with spring constant K = 2 N/m. Car 2 is 7 m from the wall at the time of collision (Car 2 is between Car 1 and the wall).

I have attached the work I have done so far, but I'm not understanding how to find x₁(t) and x₂(t), how we know Car 2 hits the wall (or moves away from it), and at what speed Car 1 travels to stay in place after link-up (given: 1 m/s, but not sure why that is).

Thank you in advance.

3 m/s
m, x₁
c)
2 X₁" = -2 X₁ + 2X₂
2
2
X₁"=-1X₁ + X₂
AX
DE
11 = K (x₂-x₁)
det (A-1)=0
=
=
ra, +
(A-21) √=0² ±2=0= (-11-
(7) 19
Hils
X (0) = a₁ - A₂
=7
A = -1
a₁ + 6₁ (t) + 2a₂
Aug. Matrix
2₂=-3
(a₁ + 2a₂)
wall ?
I m/s ?
[-+2²-2-7]
2-2
XHE
[1] (a₁ +h}) + [=1] [9₂ cos(√is t) + b₂ Sin (√5 +)
2
2 m/s
M₂ X₂= -K (X₂ X₁)
1 X₂ "1= 2X₁ -2x₂
T
T
T
X₂"= 2x₁ - 2×₂
T
2-2
= [0]
=0
MNH²,
ㅋ
[MO
21
→ (-1->)/(2-7) -(1) (2) =
2₁0 2₁₂ = -3
W₁ = 0, W₂ = √3
-a₂ cos (√3+)-by sin (√5 +)]
сос(3+)
Sin (√3+)
+262
⇒ V₁ = √₂, J₁= [1]
[ 27 V₁ =- = √₂2, 2²/= [ 21 ]
7 MALAM,
=>
010]
displacement
sprend
7 = [b₁ (f) - b₂ sin (√3+) = x ' (f) = [ 6₁ - 53 62₂ 405 (18+)
26₁ +213 12₂ c05 (√3+)
(b₁ (f) +26₂ Sin√(√3+))
Алд, тарха
RREF
2²(₂). [1-41 ] - [ ] = [101]
=
=
3
253
a₁ =0
92=0
=>
a) x² = [ { + @ (+1)(-25) 51 (28)] = [ { + + + + 51 (²+)}]}]
Sin (√3+))
353 Sin (√3+)
3+ +) (2) (-3/3) Sin (√3+)]
을 +
3-√3 Sin (√3+)
wall
6, < 1/
b₂ = -1
3√3

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