Free-Fall as 1D Motion
When you drop a ball from a window or throw a stone straight upward, the only significant force acting on it is gravity — air resistance is negligible for dense, compact objects moving at everyday speeds. This situation is called free fall, and it is one of the most elegant examples of constant-acceleration motion in all of physics. Because the acceleration never changes, every kinematic equation you learned in Unit 1 applies directly, making free fall a perfect bridge between the abstract idea of uniform acceleration and a real, observable phenomenon you can test anywhere on Earth. The key insight, established experimentally by Galileo Galilei in the late sixteenth century and later explained by Newton's law of universal gravitation, is that all objects near Earth's surface accelerate downward at the same rate regardless of their mass. This value, denoted g, is approximately 9.81 m s⁻² at sea level. A feather and a hammer fall identically in a vacuum — famously demonstrated on the Moon during Apollo 15. The independence of free-fall acceleration from mass is one of the most counter-intuitive and profound results in classical mechanics. Free fall is strictly one-dimensional (1D) motion: position, velocity, and acceleration all lie along a single vertical axis. Choosing a consistent sign convention — typically upward as positive — and substituting g = 9.81 m s⁻² as a downward (negative) acceleration transforms every free-fall problem into straightforward algebra using the four kinematic equations from Unit 1.
- Set up a sign convention
- Identify the four kinematic quantities
- Apply the kinematic equations
- Interpret the motion in three phases
- Common pitfalls
Set up a sign convention
Choose a positive direction before writing any equations. The most common choice is upward = positive, so the gravitational acceleration is a = −g = −9.81 m s⁻². Alternatively, if an object is only falling you may choose downward as positive and write a = +g. Whatever you choose, stick to it throughout the entire problem — mixing conventions is the single most common source of sign errors in free-fall problems.
Identify the four kinematic quantities
Every free-fall problem involves four variables: initial velocity u, final velocity v, displacement s (the change in vertical position, not the total distance), time t, and the constant acceleration a = ±9.81 m s⁻². Gravity supplies the acceleration automatically, so you need only two of the remaining four quantities to find everything else. Write down what is given and what is unknown before selecting an equation.
Apply the kinematic equations
The same four equations from Unit 1 apply without modification: (1) v = u + at, (2) s = ut + ½at², (3) v² = u² + 2as, and (4) s = ½(u + v)t. Free fall introduces no new mathematics — it simply specifies that a = −9.81 m s⁻² (using upward-positive). For an object thrown upward, at the peak of its trajectory v = 0, which is a powerful condition you can substitute to find maximum height or time to peak.
Interpret the motion in three phases
For an object thrown vertically upward, the motion has three natural phases: (i) upward travel — speed decreasing at 9.81 m s⁻² each second; (ii) momentary rest at the top — v = 0, but acceleration is still −9.81 m s⁻² (the object does not stop accelerating at the top!); (iii) downward fall — speed increasing at 9.81 m s⁻² each second. The trajectory is perfectly symmetric in time: the time to rise equals the time to fall back to the same height, and the speed on returning equals the launch speed.
Common pitfalls
Students often confuse displacement with distance. If a ball is thrown up 20 m and returns to the launch point, its displacement is 0 m but the total distance travelled is 40 m. The kinematic equations use displacement, not distance. Also remember that g ≈ 9.81 m s⁻² is a magnitude; the direction is always toward Earth's centre. On other planets, a different value of surface gravitational acceleration replaces 9.81 m s⁻², but the kinematic framework remains identical.
The Key Equation
- v — Final velocity (m s⁻¹): The velocity of the object at the end of the interval under consideration.
- u — Initial velocity (m s⁻¹): The velocity at the start of the interval; positive if upward (using upward-positive convention).
- a — Acceleration (m s⁻²): For free fall near Earth's surface, a = −9.81 m s⁻² (upward-positive convention); this is the only force-supplied acceleration.
- s — Displacement (m): The change in vertical position from start to end of the interval; positive upward, negative downward.
Worked Example
Problem. A student throws a ball vertically upward from ground level with an initial speed of 15.0 m s⁻¹. Using upward as positive and g = 9.81 m s⁻², find: (a) the maximum height reached, and (b) the time for the ball to return to the thrower's hand.
Answer. Maximum height ≈ 11.5 m; total flight time ≈ 3.06 s.
Origins · 1 minGalileo Galilei and how this was discovered
Galileo overturned the Aristotelian view that heavier objects fall faster by demonstrating — through careful inclined-plane experiments and logical argument — that all objects near Earth's surface undergo the same constant acceleration due to gravity, independent of mass. His kinematic analysis of free fall, published in 'Two New Sciences' (1638), introduced the concept of uniformly accelerated motion and laid the mathematical foundation for Newton's later synthesis. He was the first to treat free fall as a problem of rigorous quantitative science rather than natural philosophy. Galileo never actually dropped cannonballs from the Leaning Tower of Pisa — that story is almost certainly a legend spread by his student Vincenzo Viviani. His real evidence came from meticulously timing balls rolling down inclined planes, effectively 'diluting' gravity to slow the motion enough to measure with a water clock.
Related Concepts
- Uniform Acceleration — Free fall is the physical realisation of uniform acceleration; the kinematic equations developed for constant a apply directly, with g replacing the general symbol a.
- Kinematic Equations — The four suvat equations (v = u + at, s = ut + ½at², etc.) are the core mathematical tools used in every free-fall calculation.
- Projectile Motion — Projectile motion extends free fall to two dimensions: the vertical component is identical free-fall kinematics, while the horizontal component is uniform motion.
- Newton's Law of Universal Gravitation — Newton's law explains why g ≈ 9.81 m s⁻² at Earth's surface and how the value of free-fall acceleration depends on a planet's mass and radius.