Answer:
Three linear operators A,B, and C will satisfy the condition
.
Explanation:
According to the question we have to prove.
![[[A, B],C] + [[B,C), A] + [[C, A], B] = 0](https://tex.z-dn.net/?f=%5B%5BA%2C%20B%5D%2CC%5D%20%2B%20%5B%5BB%2CC%29%2C%20A%5D%20%2B%20%5B%5BC%2C%20A%5D%2C%20B%5D%20%3D%200)
Now taking Left hand side of the equation and solve.
![[[A, B],C] + [[B,C), A] + [[C, A], B]](https://tex.z-dn.net/?f=%5B%5BA%2C%20B%5D%2CC%5D%20%2B%20%5B%5BB%2CC%29%2C%20A%5D%20%2B%20%5B%5BC%2C%20A%5D%2C%20B%5D)
Now use commutator property on it as,
![=[A,B] C-C[A,B]+[B,C]A-A[B,C]+[C,A]B-B[C,A]\\=(AB-BA)C-C(AB-BA)+(BC-CA)A-A(BC-CB)+(CA-AC)B-B(CA-AC)\\=ABC-BAC-CAB+CBA+BCA-CAB-ABC+ACB+CAB-ACB-BCA+BAC\\=0](https://tex.z-dn.net/?f=%3D%5BA%2CB%5D%20C-C%5BA%2CB%5D%2B%5BB%2CC%5DA-A%5BB%2CC%5D%2B%5BC%2CA%5DB-B%5BC%2CA%5D%5C%5C%3D%28AB-BA%29C-C%28AB-BA%29%2B%28BC-CA%29A-A%28BC-CB%29%2B%28CA-AC%29B-B%28CA-AC%29%5C%5C%3DABC-BAC-CAB%2BCBA%2BBCA-CAB-ABC%2BACB%2BCAB-ACB-BCA%2BBAC%5C%5C%3D0)
Therefore, it is proved that
.
We use the formula,

Here, h is the variable represents the height of the flare in feet when it returns to the sea so, h = 0 and u is the initial velocity of the flare, in feet per second and its value of 192 ft/sec.
Substituting these values in above equation, we get
.
Here, t= 0 neglect because it is the time when the flare is launched.
Thus, flare return to the sea in 12 s.
Explanation:
Load=800N
Effort=200N
1. Mechanical Advantage = LOAD/EFFORT
= 800N/200N
= 4
2 Velocity Ratio = no. Of pulleys =5
3. Efficiency = Mechanical advantage / velocity ratio × 100%
= (4/5)×100%
=80%
4. output work= load×load distance
= 800N × 5m
= 4 × 1000J
5. Efficiency = (output work/input work) ×100%
Or, 80% = (4000J/input work) ×100%
Or, 80%/100% = 4000J/inputwork
Or, 4/5 = 4000J/inputwork
Or, input work =4000J × 5/4
Input work = 5×1000J
I hope it helped! ;-)
Answer:
v0 = 25 m/s
vf = 0 m/s
a = -9.80 m/s^2
change in x = 31.89m
but that's only 1/3 of the hight, so i time it by 3 to get 96m