Coulomb's wedge theory — the direction of movement of the active earth-pressure wedge relative to the retaining wall is:

Introduction / Context:
Retaining walls supporting backfill are subjected to earth pressures that depend on the wall movement. Under active conditions, the wall yields sufficiently away from the soil to mobilize a lower lateral pressure. Coulomb's wedge theory models the backfill behind the wall as a potential sliding wedge bounded by a planar failure surface and evaluates equilibrium to find the extreme (active/passive) pressures.

Given Data / Assumptions:

• Backfill approximated as a rigid wedge with planar failure surface.
• Active condition: wall moves away from backfill.
• Frictional soil with angle of shearing resistance phi.
• Resultant of soil weight and reactions drives motion along the failure plane.

Concept / Approach:

Under active conditions, the soil mass seeks to expand laterally, and the failure wedge tends to slide downwards and outwards along a slip plane inclined to the horizontal. The retaining wall moves away from the backfill, and the wedge moves toward the wall’s back face while descending due to gravity, consistent with Coulomb’s free-body analysis.

Step-by-Step Solution:

Verification / Alternative check:

Compare with passive state: for passive pressure, the wall pushes into the soil and the wedge tends to move upward and inward, opposite to the active case — reinforcing the active-direction conclusion.

Why Other Options Are Wrong:

Up and inwards — characteristic of passive, not active conditions.

Horizontal movements parallel to base neglect gravitational component along the slip plane.

Pure rotation about the heel is not Coulomb’s assumed wedge mechanism.

Common Pitfalls:

Confusing active with passive wedge kinematics; ignoring wall friction and backfill slope effects when applying Coulomb formulas.

Final Answer:

down and outwards on a slip surface