Finite Element Analysis 1D Problem

Question

I have two 1D problems:
A) A rocket that goes upwards with acceleration a, upwards, so positive, using it's engine thrust. So it has a field force:
fx1 = a*ρ (ρ:density) and nodal forces: Fb1=Fb2=A*fx1*L/2
B) A 1D bar in the x-direction that is fixed to point A (left) and goes to the right with an acelaration to the right of 10g (g=9.81 m/s^2). So it has a field force: fx2 = 10g*ρ (ρ:density) and nodal forces: Fb1=Fb2=A*fx2*L/2

Which one is the correct answer (Use d'alembert or not) :

1) fx1 = - a*ρ (negative) downwards and fx2 = 10g*ρ (positive) to the right

2) fx2 = - a*ρ (negative) downwards and fx2 = -10g*ρ (negative) to the left

3) fx1 = a*ρ (positive) upwards and fx2 = 10g*ρ (positive) to the right

4) fx1 =  a*ρ (positive) upwards and fx2 = -10g*ρ (negative) to the left

Answer 1

For Problem A (Rocket with upward acceleration): The rocket accelerates upward with acceleration 'a' and the engine provides thrust. The body force is f=a*p (ρ=density) due to acceleration. Since it goes upward (positive), and the engine thrust is upward, the nodal forces would be Fb1=Fb2=A*fx1*L/2 where fx1 is positive (upward force). For Problem B (1D bar with horizontal acceleration): The bar accelerates to the right with g=9.81 m/s^2 (given as 10g). The body force is fx2 = 10g*p (positive to the right). The correct answer considering both problems' force directions is option 3) fx1 = a*p (positive) upwards and fx2 = 10g*p (positive) to the right. This correctly represents both problems where the accelerations create positive body forces in their respective directions of motion.