Motion Vertical under Gravity

Kinematic study of vertical rectilinear motion under gravity, free fall, launches, and depth from fall time.

Conceptual context

Vertical motion under gravity studies objects that rise or fall near the Earth's surface when the dominant action is gravity. It is a case of UARM, but it deserves its own treatment because the vertical axis forces careful thinking about sign, height, reference point and the physical meaning of velocity.

Three situations are very common in this topic: free fall from rest, vertical launch upwards and vertical launch downwards. In all of them gravitational acceleration points downwards. What changes from one problem to another is not gravity, but the initial velocity, the initial vertical position, the chosen origin and the question being asked.

The important historical idea is that, without air resistance, all bodies fall with the same acceleration regardless of their mass. Galileo turned this intuition into an experimental law, and the hammer and feather demonstration on the Moon showed it visually: with the air removed, both objects fall together.

🟢 Essential level

The first reading the student must build is qualitative. If you throw a ball upwards, it keeps rising for a while, but each moment more slowly. At the highest point it stops for an instant and then starts to fall. Throughout the whole process, gravity has been acting downwards.

The second reading concerns signs. If we choose the vertical axis positive upwards, a velocity upwards is positive and a velocity downwards is negative. Gravity, on the other hand, enters with a downward sense. This convention is not an algebraic detail: it is how we translate into mathematical language what it means to rise, fall or stop.

The third reading concerns position. The vertical position indicates where the object is on the vertical axis; the initial vertical position indicates where it was at the start; the height is usually a positive distance measured from a reference level. It is worth not mixing these ideas: a height is a distance from a level, whereas a vertical position can be positive, negative or zero depending on the chosen origin.

With these three ideas one can already reason before calculating. If an object is released from rest, its velocity is initially zero and then grows in the downward sense. If it is launched upwards, it first loses upward velocity, then reaches the maximum height and finally descends. If it is launched downwards, it already starts with a downward velocity and gravity increases that tendency.

🔵 Formal level

Basic formulas

Six formal relations are used in this topic. The vertical position-time equation gives the vertical position from the initial position, the initial velocity, the time and gravity. The vertical velocity equation describes how the vertical velocity changes with time. The vertical Torricelli equation connects velocity and position without passing through time, via the net vertical displacement between both instants. The three auxiliary relations give the impact time from a height, the time to rise and the maximum height in ideal vertical launches.

Vertical position-time equation

Vertical velocity-time equation

Vertical Torricelli's equation

Real student questions

Why does a sheet of paper fall more slowly than a stone? Because of air resistance. If you crumple the sheet, its behaviour is closer to that of a compact body.

Is gravity 9.8 or 10? For standard calculations 9.8 m/s² is used. For quick estimates, 10 m/s² is usually enough. What matters is to state which value has been used.

Why does the velocity become negative? Because the sign depends on the chosen axis. If upwards is positive, falling is negative.

What happens on the Moon? The same ideas are used, but with a smaller gravity. That is why objects fall more slowly and jumps reach greater heights.

Final synthesis

Vertical motion under gravity describes the position and velocity of an object when gravity is the dominant force. Its main difficulty lies in choosing the axis correctly, interpreting signs, distinguishing height from position and recognising when the ideal model is no longer valid. Before calculating, one should be able to state whether the object rises, falls, stops or reaches the ground.