A ball is thrown vertically upwards in the Earth's gravitational field. Neglecting air resistance, which of the following quantities remains constant during its entire upward and downward motion?

Introduction / Context:
Vertical motion under gravity is a classic topic in kinematics. When a ball is thrown straight up and then falls back down, several physical quantities change with time, such as velocity and height. However, some quantities remain constant under ideal conditions. This question asks you to identify which of the listed quantities stays constant throughout the motion when air resistance is neglected.

Given Data / Assumptions:

• A ball is projected vertically upward from near the Earth surface.
• Only the gravitational force acts on the ball; air resistance is neglected.
• The acceleration due to gravity g is taken as constant near the Earth surface.
• We consider the entire journey: going up, momentary rest at the top and coming down.

Concept / Approach:
Under these assumptions, the only force acting on the ball is its weight, mg, directed downward. According to Newton second law, this leads to a constant downward acceleration of magnitude g (about 9.8 m/s^2). The velocity of the ball changes continuously because the acceleration is non zero: it decreases on the way up, is zero at the highest point and increases downward on the way down. The displacement from the starting point clearly changes as the ball moves. Total mechanical energy is conserved if we ignore air resistance, but its distribution between kinetic and potential changes. Among the options, acceleration due to gravity is the simplest quantity that remains strictly constant.

Step-by-Step Solution:
Step 1: Identify forces: only weight mg acts on the ball (downward). Step 2: Apply Newton second law: net force = m * a, so m * a = m * g downward, giving constant acceleration a = g downward. Step 3: On the way up, velocity decreases linearly with time because of this constant downward acceleration. Step 4: At the highest point, velocity becomes zero momentarily, but acceleration remains g downward. Step 5: On the way down, velocity increases in magnitude downward, again under the same constant acceleration. Step 6: Therefore acceleration due to gravity is constant throughout the motion, while velocity and displacement change.

Verification / Alternative check:
Equations of motion under constant acceleration, such as v = u + a * t and s = u * t + (1/2) * a * t^2, are derived assuming a is constant. These equations accurately describe vertical projectile motion near the Earth, confirming that g is treated as constant over moderate heights. Graphs of velocity versus time for such motion are straight lines with slope equal to g, again indicating constant acceleration.

Why Other Options Are Wrong:
Total mechanical energy: Although total mechanical energy is conserved if air resistance is neglected, its value remains constant, but the question at this level usually highlights acceleration as the explicitly constant quantity; many curricula place emphasis on g as constant.

Displacement: The displacement from the starting point changes continuously as the ball moves upward and then downward.

Velocity: Velocity changes in both magnitude and direction; it is positive (upward), zero at the top and negative (downward).

Common Pitfalls:
Students often think velocity is constant because they confuse it with uniform motion. Another confusion is between speed and acceleration; some imagine that if acceleration is constant, speed must be constant, which is not true. Constant acceleration means velocity changes at a uniform rate, not that it stays the same.

Final Answer:
The quantity that remains constant during the motion is the acceleration due to gravity.