Upward & Downward Motion Under Gravity
When you throw a ball straight up into the air, something remarkable happens: gravity continuously decelerates it, brings it to a momentary stop, then accelerates it back downward — all without any engine or push after the initial throw. This entire journey is governed by one constant force: Earth's gravitational pull, which gives every freely falling object an acceleration of approximately 9.8 m s⁻² directed downward. Understanding this motion requires careful attention to direction, because velocity and acceleration can point opposite ways simultaneously. The key mathematical tool is sign convention — the deliberate choice of which direction (up or down) is positive. Once chosen, you must apply it consistently throughout a problem. If you define upward as positive, then the gravitational acceleration g is entered as −9.8 m s⁻² in every equation. A ball moving upward has positive velocity; the same ball moving downward has negative velocity. The acceleration, however, is always −9.8 m s⁻² regardless of the ball's direction of travel. This is one of the most common sources of error in exam answers. A beautiful consequence of constant acceleration under gravity is time symmetry: the time a projectile takes to reach its highest point equals the time it takes to fall back to the same height. Similarly, the speed at which it leaves a point on the way up equals the speed at which it passes that same point on the way down. These symmetries simplify many calculations and serve as powerful checks on your working.
- Choosing a Sign Convention
- The Three Phases of Upward Projection
- Velocity Reversal at the Apex
- Time Symmetry Explained
- Displacement vs Distance
Choosing a Sign Convention
Before writing a single equation, declare your positive direction. The most common choice for vertical problems is upward = positive. Write this at the top of your working. Every vector quantity — displacement, velocity, acceleration — must use this convention. Gravitational acceleration is then a = −g = −9.8 m s⁻². If instead you choose downward = positive (useful when objects only fall), then a = +g = +9.8 m s⁻². Neither convention is wrong; inconsistency is the only error.
The Three Phases of Upward Projection
Phase 1 — Upward journey: the object has positive velocity (if upward is positive) but negative acceleration. The velocity decreases in magnitude at 9.8 m s⁻¹ every second. Phase 2 — Apex (highest point): velocity is instantaneously zero, but acceleration is still −9.8 m s⁻². The object does not 'pause'; it reverses direction in an instant. Phase 3 — Downward journey: velocity is now negative and increasing in magnitude. The object speeds up as it falls. These phases share a single set of kinematic equations — no new physics is needed at the turnaround.
Velocity Reversal at the Apex
A common misconception is that acceleration is zero at the highest point because velocity is zero. This is false. Acceleration is the rate of change of velocity, not velocity itself. At the apex, velocity changes from a small positive value to zero to a small negative value — it is changing most smoothly through zero. Gravity never switches off. Think of it like a car braking to a stop: the deceleration exists right up to (and beyond) the moment of zero speed.
Time Symmetry Explained
Because acceleration is constant throughout the flight, the equations are symmetric in time about the apex. If the ball takes time t_up to reach the top, it takes exactly t_up to return to the launch height. Mathematically, this follows from the velocity equation v = u + at: at the apex, v = 0, so t_up = u/g. On the way down, starting from rest, it reaches −u after time u/g. The symmetry also means |speed at any height on the way up| = |speed at the same height on the way down|.
Displacement vs Distance
Be careful to distinguish displacement (a vector, can be negative) from distance (always positive). If you throw a ball upward and it returns to your hand, its displacement is zero but the distance travelled is 2 × (maximum height). Kinematic equations use displacement, so if the ball lands at the same point from which it was launched, you can set s = 0 directly — a powerful shortcut that gives the total flight time in one step.
The Key Equation
- v — Final velocity (m s⁻¹): Velocity at the end of the time interval; negative if moving downward (upward-positive convention).
- u — Initial velocity (m s⁻¹): Velocity at the start of the time interval; positive if moving upward.
- a — Acceleration (m s⁻²): For free fall with upward positive, a = −9.8 m s⁻². Constant throughout the motion.
- t — Time (s): Duration of the interval being analysed; always positive.
- s — Displacement (m): Change in position from start to end of interval; positive = upward, negative = downward.
Worked Example
Problem. A tennis ball is thrown vertically upward from ground level with an initial speed of 19.6 m s⁻¹. Taking upward as positive and g = 9.8 m s⁻², find: (a) the time to reach maximum height, (b) the maximum height reached, and (c) the total time of flight before returning to ground level.
Answer. Time to apex = 2.0 s; Maximum height = 19.6 m; Total flight time = 4.0 s.
Origins · 1 minGalileo Galilei and how this was discovered
Galileo was the first to establish through systematic experiment that all objects fall with the same constant acceleration regardless of mass, and that vertical motion follows precise mathematical rules. His inclined-plane experiments allowed him to measure the time-squared relationship between distance and time — equivalent to our modern kinematic equations. He recognised that a projectile's vertical and horizontal motions are independent, laying the foundation for analysing free fall and vertical projection quantitatively. Galileo is said to have dropped two cannonballs of different masses from the Leaning Tower of Pisa to demonstrate equal-acceleration free fall — though historians debate whether this dramatic experiment actually occurred or was a thought experiment he described in writing.
Related Concepts
- Gravitational Acceleration — Free fall is defined by the constant gravitational acceleration g; understanding its origin and value is essential before analysing vertical motion.
- Projectile Motion — Projectile motion extends vertical free-fall analysis to include horizontal motion, treating the vertical component with exactly the same sign-convention methods.
- Newton's Law of Gravitation — The constant downward acceleration in free fall arises directly from Newton's universal law of gravitation acting between Earth and the falling object.
- Kinematic Equations — The three suvat equations used throughout vertical motion problems are derived from the definition of constant acceleration and must be mastered first.