A ball is thrown up in the air. It reaches a maximum height and then returns…

For any object moving freely under gravity near Earth's surface (ignoring air resistance), the gravitational acceleration g and the gravitational force F = mg acting on it are constant vectors — fixed in magnitude and always directed vertically downward — at every instant of the motion, whether the object is moving up or down. Displacement (position measured from the start) and work done by gravity over a given leg of the journey (which depends on the direction of the ball's overall displacement across that leg relative to the force), by contrast, both change with the direction of travel.

The question specifies that the quantity must remain unchanged in value at every instant throughout the ball's ENTIRE flight — from the moment it leaves the thrower's hand to the moment it returns — not merely whether two instantaneous values happen to match at a pair of symmetric instants, such as the same height crossed once on the way up and once on the way down.

Throughout both the rise and the fall, gravity is the only force acting (air resistance ignored), so the acceleration stays equal to g, directed downward, at every instant — it never changes whether the ball is going up or coming down. The force of gravity, F = mg, depends only on the ball's mass and g, neither of which changes during the flight, so this force is likewise constant at every instant. Displacement, however, is measured from the throw point: it grows from 0 up to the maximum height H during the rise, then shrinks back from H to 0 during the fall — its value is different at nearly every instant of the flight (it only momentarily coincides at a shared height crossed twice), so it is not unchanged throughout the whole journey. Work done by gravity, considered over each leg of the journey as a whole, is negative over the entire upward leg, W_up = -mgH (gravity opposes the ball's overall upward displacement on that leg) and positive over the entire downward leg, W_down = +mgH (gravity acts along the ball's overall downward displacement on that leg) — these two phase totals carry opposite signs, so work done is not unchanged throughout the whole journey either.

• Acceleration — stays equal to g, directed downward, at every instant of the rise and the fall; unchanged throughout the entire flight.

• Force of gravity — stays equal to mg, directed downward, at every instant of the rise and the fall; unchanged throughout the entire flight.

• Displacement — increases from 0 to H while rising, then decreases from H back to 0 while falling; a different value at nearly every instant, not unchanged throughout the flight.

• Work done by gravity — negative over the whole upward leg, positive over the whole downward leg; the two phase totals carry opposite signs, so it is not unchanged throughout the flight.

Since acceleration and the force of gravity remain unchanged at every instant of the ball's entire upward-and-downward flight, while displacement and work done both vary across it, the quantities with the same value throughout the motion are acceleration and the force of gravity.