Which of the following equations is dimensionally consistent?
A. a3 = 1.2 (v/t)3 + 0.2 x3/t6 + 1.5 v4/x3
B. a3 = 1.2 (v/t)3 + 0.2 x2/t6 +3 v6/x3
C. a3 = 1.2 (v/t)3 + 0.2 x3/t6 - 1.5 v6/x3
D. a3 = 1.5 (v/t)2 + 0.2 x3/t6 - 1.5 v6/x3
To determine which equation is dimensionally consistent, we need to check if the dimensions of the terms on both sides of the equation match.
Let's analyze each term in the given equations:
Term 1: a3 (acceleration cubed)
Term 2: 1.2 (v/t)3 (velocity divided by time, cubed)
Term 3: 0.2 x3/t6 (position cubed divided by time to the power of 6)
Term 4: 1.5 v4/x3 (velocity to the power of 4 divided by position cubed)
Term 5: 1.2 (v/t)3 (velocity divided by time, cubed)
Term 6: 0.2 x2/t6 (position squared divided by time to the power of 6)
Term 7: 3 v6/x3 (velocity to the power of 6 divided by position cubed)
Term 8: 1.2 (v/t)3 (velocity divided by time, cubed)
Term 9: 0.2 x3/t6 (position cubed divided by time to the power of 6)
Term 10: -1.5 v6/x3 (negative velocity to the power of 6 divided by position cubed)
Term 11: 1.5 (v/t)2 (velocity divided by time, squared)
Term 12: 0.2 x3/t6 (position cubed divided by time to the power of 6)
Term 13: -1.5 v6/x3 (negative velocity to the power of 6 divided by position cubed)
Now, let's break down the dimensions of each term:
Term 1: a3 (acceleration cubed) -> [L/T^2]^3 = [L^3/T^6]
Term 2: 1.2 (v/t)3 (velocity divided by time, cubed) -> [L/T]^3 = [L^3/T^3]
Term 3: 0.2 x3/t6 (position cubed divided by time to the power of 6) -> [L^3/T^6]
Term 4: 1.5 v4/x3 (velocity to the power of 4 divided by position cubed) -> [L/T]^4 / [L^3] = [L/T^4]
Term 5: 1.2 (v/t)3 (velocity divided by time, cubed) -> [L/T]^3 = [L^3/T^3]
Term 6: 0.2 x2/t6 (position squared divided by time to the power of 6) -> [L^2/T^6]
Term 7: 3 v6/x3 (velocity to the power of 6 divided by position cubed) -> [L/T]^6 / [L^3] = [L^3/T^6]
Term 8: 1.2 (v/t)3 (velocity divided by time, cubed) -> [L/T]^3 = [L^3/T^3]
Term 9: 0.2 x3/t6 (position cubed divided by time to the power of 6) -> [L^3/T^6]
Term 10: -1.5 v6/x3 (negative velocity to the power of 6 divided by position cubed) -> [L/T]^6 / [L^3] = [L^3/T^6]
Term 11: 1.5 (v/t)2 (velocity divided by time, squared) -> [L/T]^2 = [L^2/T^2]
Term 12: 0.2 x3/t6 (position cubed divided by time to the power of 6) -> [L^3/T^6]
Term 13: -1.5 v6/x3 (negative velocity to the power of 6 divided by position cubed) -> [L/T]^6 / [L^3] = [L^3/T^6]
Now, let's check the equations for dimensional consistency:
A. a3 = 1.2 (v/t)3 + 0.2 x3/t6 + 1.5 v4/x3
-> [L^3/T^6] = [L^3/T^3] + [L^3/T^6] + [L^3/T^4]
The dimensions on both sides are not consistent, so equation A is NOT dimensionally consistent.
B. a3 = 1.2 (v/t)3 + 0.2 x2/t6 +3 v6/x3
-> [L^3/T^6] = [L^3/T^3] + [L^2/T^6] + [L^3/T^6]
The dimensions on both sides are not consistent, so equation B is NOT dimensionally consistent.
C. a3 = 1.2 (v/t)3 + 0.2 x3/t6 - 1.5 v6/x3
-> [L^3/T^6] = [L^3/T^3] + [L^3/T^6] - [L^3/T^6]
The dimensions on both sides are consistent, so equation C is dimensionally consistent.
D. a3 = 1.5 (v/t)2 + 0.2 x3/t6 - 1.5 v^6/x3
-> [L^3/T^6] = [L^2/T^2] + [L^3/T^6] - [L^3/T^6]
The dimensions on both sides are not consistent, so equation D is NOT dimensionally consistent.
Therefore, out of the given equations, only equation C is dimensionally consistent.