A tank is partially filled with a liquid and is given a uniform horizontal acceleration. How does the free surface of the liquid adjust itself inside the tank?

Introduction / Context:
When a container with liquid is accelerated horizontally, the free surface of the liquid does not remain horizontal. Instead, it tilts because each fluid element experiences an effective acceleration that is the vector sum of gravitational acceleration and the imposed horizontal acceleration. This question from fluid mechanics tests your understanding of how the free surface adjusts under uniform horizontal acceleration. Knowing this behaviour is important for analysing liquid levels in moving vehicles, tanks on trucks, and many engineering applications.

Given Data / Assumptions:
• The tank is partially filled with a liquid. • The tank experiences a uniform horizontal acceleration. • The acceleration is in a fixed horizontal direction, which we call the forward direction of motion. • The liquid is considered incompressible and has a free surface open to the atmosphere.

Concept / Approach:
In a non accelerating frame, the free surface of a liquid at rest is always horizontal because pressure at any depth is hydrostatic and depends only on vertical depth. When the container is accelerated horizontally with acceleration a, an observer moving with the container can treat the situation as if there is an additional pseudo acceleration acting on the fluid in the opposite direction. The effective acceleration is the vector sum of gravity g acting downward and this horizontal acceleration a. The free surface of the liquid aligns itself perpendicular to the direction of the resultant acceleration. Because the resultant acceleration tilts backward relative to the direction of motion, the free surface tilts such that it falls at the front (direction of acceleration) and rises at the back.

Step-by-Step Solution:
Step 1: Consider the tank accelerating horizontally to the right with acceleration a. Step 2: In the accelerating frame of reference attached to the tank, the liquid experiences gravity g downward and an equivalent pseudo acceleration a to the left. Step 3: The resultant of g and a points downward and backward relative to the direction of motion. Step 4: The free surface of a liquid is always perpendicular to the resultant acceleration vector. Step 5: Therefore, the free surface tilts so that it is higher at the back side and lower at the front side in the direction of motion.

Verification / Alternative check:
A simple everyday observation can confirm this behaviour. If you hold a cup of water and suddenly accelerate forward, the water appears to slosh backward and rise more near the rear side of the cup while becoming lower at the front. This is consistent with the concept of effective acceleration. In engineering textbooks, the slope of the free surface is often given by tan(theta) = a / g, where theta is the angle that the free surface makes with the horizontal. The sign of this slope clearly shows that the surface drops in the direction of motion and rises on the opposite side, matching the description in the correct option.

Why Other Options Are Wrong:
Option b: The liquid surface does not fall only at the center; instead it forms a tilted plane surface, not a dip at the center. Option c: The free surface cannot remain perfectly horizontal under horizontal acceleration because pressure distribution is no longer purely hydrostatic. Option d: This reverses the correct behaviour, claiming that the surface rises on the forward side and falls on the back side, which contradicts both theory and observation. Option e: A vertical plane surface would imply an infinite slope, which does not occur in this situation.

Common Pitfalls:
Many learners incorrectly imagine the liquid lagging behind and think the surface rises at the front, but this intuition ignores the direction of the effective acceleration. Another confusion comes from mixing frames of reference. It is easier to analyse the problem in the accelerating frame and imagine a pseudo acceleration acting opposite to the motion. Remember that the free surface must always be perpendicular to the net acceleration vector acting on the fluid. Drawing a simple vector diagram of g and a helps avoid conceptual errors.

Final Answer:
The free surface falls on the forward side and rises on the back side, so the correct statement is: The liquid surface falls on the forward side in the direction of motion and rises on the back side of the tank.