Range, Time of Flight & Maximum Height
When you kick a football or throw a javelin, the object follows a curved path through the air. This curved trajectory is called projectile motion, and it arises because gravity acts downward whilst the object retains whatever horizontal velocity it was given at launch. Three numbers completely describe the flight: the maximum height reached, the time the object spends airborne, and the horizontal distance — called the range — covered before landing. Understanding these quantities lets engineers design catapults, athletes optimise throwing angles, and physicists model everything from cannonballs to spacecraft re-entry. The secret to analysing projectile motion is recognising that horizontal and vertical motions are entirely independent. Horizontally, there is no acceleration (ignoring air resistance), so the object moves at a constant speed throughout. Vertically, gravity provides a constant downward acceleration of g ≈ 9.81 m s⁻². By treating these two directions separately and then combining their effects, we can derive exact formulae for maximum height, time of flight, and range — all as functions of the initial speed and the launch angle above the horizontal. These formulae are not arbitrary — each one is derived directly from the equations of uniform acceleration (the SUVAT equations) applied along one axis at a time. Once derived, they reveal beautiful patterns: for instance, complementary angles such as 30° and 60° give identical ranges, and the maximum range on flat ground always occurs at exactly 45°. These elegant results make projectile motion one of the most satisfying topics in classical mechanics.
- Splitting initial velocity into components
- Deriving time of flight T
- Deriving maximum height H
- Deriving horizontal range R
- Symmetry and complementary angles
Splitting initial velocity into components
A projectile launched at speed u and angle θ above the horizontal has an initial horizontal component uₓ = u cos θ and an initial vertical component uᵧ = u sin θ. Think of it like a lift and a walk happening simultaneously: the vertical component fights gravity, whilst the horizontal component carries the object forward at constant speed.
Deriving time of flight T
At the top of the trajectory the vertical velocity is zero. Using v = u + at vertically: 0 = u sin θ − g t_top, so t_top = (u sin θ)/g. The projectile then takes an equal time to fall back to the same height (by symmetry), so the total time of flight is T = 2u sin θ / g. This doubling is the same reason a ball thrown upward takes as long to fall as it took to rise.
Deriving maximum height H
Using v² = u² + 2as vertically, with final vertical velocity = 0 at the peak: 0 = (u sin θ)² − 2g H. Rearranging gives H = u² sin²θ / (2g). Notice that H depends on sin²θ, so it is maximised when θ = 90° (straight up) — which makes intuitive sense, even if that produces zero range.
Deriving horizontal range R
Since there is no horizontal acceleration, range = horizontal speed × time of flight: R = (u cos θ) × T = (u cos θ) × (2u sin θ / g) = u² sin 2θ / g. This uses the double-angle identity 2 sin θ cos θ = sin 2θ. Because sin 2θ is maximised when 2θ = 90°, i.e. θ = 45°, the maximum possible range on flat ground is R_max = u²/g.
Symmetry and complementary angles
Because R = u² sin 2θ / g and sin 2θ = sin(180° − 2θ), pairs of angles that add to 90° (such as 30° & 60°, or 20° & 70°) produce the same range. This is why a shot-putter does not need to choose exactly 45° — a slightly different angle still gives a near-maximum range. In practice, air resistance and the height of release modify these ideal results.
The Key Equation
- R — Horizontal range (metres (m)): The total horizontal distance travelled from launch to landing on the same horizontal level.
- H — Maximum height (metres (m)): The greatest vertical distance above the launch point, reached when the vertical velocity becomes zero.
- T — Time of flight (seconds (s)): The total time the projectile remains airborne, from launch until it returns to the original height.
- u — Initial speed (metres per second (m s⁻¹)): The magnitude of the velocity at the moment of launch.
- \theta — Launch angle (degrees or radians): The angle between the initial velocity vector and the horizontal.
- g — Gravitational acceleration (metres per second squared (m s⁻²)): The constant downward acceleration due to gravity, approximately 9.81 m s⁻² near Earth's surface.
Worked Example
Problem. A tennis ball is struck at an initial speed of 20 m s⁻¹ at an angle of 30° above the horizontal from ground level. Calculate (a) the time of flight, (b) the maximum height, and (c) the horizontal range. Take g = 9.81 m s⁻².
Answer. Time of flight T ≈ 2.04 s; Maximum height H ≈ 5.10 m; Horizontal range R ≈ 35.3 m.
Origins · 1 minGalileo Galilei and how this was discovered
Galileo was the first scientist to correctly analyse projectile motion, demonstrating in his book 'Two New Sciences' (1638) that the horizontal and vertical components of motion are independent. He showed that the trajectory is a parabola and derived the dependence of range on launch angle, establishing the foundation upon which all modern projectile equations rest. Galileo reportedly carried out early experiments on motion by rolling balls down inclined ramps rather than dropping them, slowing the action enough to measure with a water-clock — an ingenious workaround before precision stopwatches existed.
Related Concepts
- Velocity and Acceleration — Range, height, and time of flight are all derived using the definitions of velocity and constant acceleration — the building blocks of all projectile calculations.
- Equations of Uniform Acceleration (SUVAT) — The SUVAT equations are applied separately to horizontal and vertical directions to derive every projectile formula; mastering them is essential.
- Vectors and Vector Components — Resolving initial velocity into horizontal and vertical components using trigonometry is the first and most critical step in every projectile problem.
- Free Fall and Acceleration Due to Gravity — Projectile motion is essentially free fall combined with constant horizontal motion; understanding g and free fall underpins the vertical part of all projectile analysis.