NCEA Level 2 · Year 12 · Physics · 2 minutes

Why a thrown ball curves: projectile motion

Two independent motions, one shared time: that is projectile motion.

Key terms

projectile
An object moving through the air under the pull of gravity, like a kicked ball.
parabola
The curved path that results from combining steady sideways motion with a fall that keeps speeding up.
independent motions
Horizontal and vertical motion do not affect each other.
horizontal velocity
The sideways speed, which stays constant because no force acts horizontally when air resistance is ignored.
vertical acceleration
The change in up and down speed, 9.8 metres per second squared downwards because of gravity.
components
The horizontal and vertical parts of a launch velocity, found using cosine and sine.

Transcript

0:00

Kick a rugby ball or throw a cricket ball, and it follows a curve. Before Galileo, even the experts drew that curve wrong. Getting it right takes one big idea.

0:10

In fifteen thirty seven, a famous book on gunnery drew a cannonball's path in three parts: shooting straight out, curving round, then dropping straight down. Sensible, but gravity has other plans.

0:23

A century later, Galileo saw it differently. In his last book, in sixteen thirty eight, he treated a projectile as two motions happening at once: a steady sideways motion, and a fall that keeps speeding up. Put them together, and you get a parabola.

0:39

The key word for your exam: independent. The horizontal and vertical motions don't affect each other. Kick one ball sideways off a cliff and drop another at the same moment. Ignoring air resistance, they hit the ground together.

0:52

So, with no air resistance, there are two rules. Horizontally, there's no force, so the velocity stays constant. Vertically, gravity pulls down, so the acceleration is nine point eight metres per second squared, downwards, the whole time. At the very top, the vertical velocity is zero for an instant, but gravity doesn't take a break.

1:14

Let's try one. A ball is kicked horizontally at fifteen metres per second off a cliff nineteen point six metres high. How far out does it land? Start vertically. It begins with no vertical velocity, so the distance fallen equals half times acceleration times time squared. Nineteen point six equals four point nine times time squared, so the time is two seconds.

1:37

Now horizontally. Distance equals velocity times time: fifteen times two. It lands thirty metres out. Time is the bridge: it's the one thing both directions share.

1:50

For a kick at an angle, like a conversion, first split the launch velocity into horizontal and vertical components, using cosine and sine. Then it's the same two rules.

2:01

So, three things to remember. One: horizontal and vertical motion are independent. Two: horizontal velocity is constant, and vertical acceleration is nine point eight downwards. Three: find the time from one direction, and use it in the other. Galileo worked that out nearly four hundred years ago. Now it's your turn.

Sources

Every fact in the video, and where it comes from.

  1. 1
    arxiv.org/pdf/1804.00716
    • Niccolò Tartaglia's Nova scientia (1537), a book applying mathematics to artillery fire, drew projectile paths in three parts: a 'violent' (near straight) motion, a curved 'mixed' motion, then a 'natural' straight-down fall.
  2. 2
    mathshistory.st-andrews.ac.uk/Biographies/Tartaglia
    • Niccolò Tartaglia's Nova scientia (1537), a book applying mathematics to artillery fire, drew projectile paths in three parts: a 'violent' (near straight) motion, a curved 'mixed' motion, then a 'natural' straight-down fall.
  3. 3
    commons.wikimedia.org/wiki/File:Projectile_trajectories_from_Tartagl…
    • Niccolò Tartaglia's Nova scientia (1537), a book applying mathematics to artillery fire, drew projectile paths in three parts: a 'violent' (near straight) motion, a curved 'mixed' motion, then a 'natural' straight-down fall.
  4. 4
    en.wikipedia.org/wiki/Two_New_Sciences
    • Galileo's Two New Sciences (Discorsi e dimostrazioni matematiche intorno a due nuove scienze) was published in 1638 in Leiden; it was his final book.
    • In the Fourth Day of Two New Sciences, Galileo shows that a projectile carried by a uniform horizontal motion compounded with a naturally accelerated vertical motion describes a semi-parabola.
  5. 5
    galileoandeinstein.phys.virginia.edu/tns244.htm
    • In the Fourth Day of Two New Sciences, Galileo shows that a projectile carried by a uniform horizontal motion compounded with a naturally accelerated vertical motion describes a semi-parabola.
  6. 6
    openstax.org/books/university-physics-volume-1/pages/4-3-pr…
    • Motions along perpendicular axes are independent and can be analysed separately; without air resistance the horizontal velocity is constant; the trajectory is a parabola; at the highest point the vertical velocity is zero.
    • Near Earth's surface the acceleration due to gravity is about 9.8 m/s² downward (standard gravity 9.80665 m/s²), and it acts throughout the flight, including at the top of the path.
    • Ignoring air resistance, a ball launched horizontally and a ball dropped from the same height at the same moment reach the ground at the same time, because both start with zero vertical velocity and have the same vertical acceleration.
    • Worked example (calculated): horizontal launch at 15 m/s from 19.6 m. Vertical: d = v_i t + ½ a t² with v_i = 0 → 19.6 = 4.9 t² → t = 2.0 s. Horizontal: d = v t = 15 × 2.0 = 30 m.
    • For a launch at an angle, the initial velocity is resolved into components: v_x = v cos θ, v_y = v sin θ.
  7. 7
    physics.nist.gov/cgi-bin/cuu/Value
    • Near Earth's surface the acceleration due to gravity is about 9.8 m/s² downward (standard gravity 9.80665 m/s²), and it acts throughout the flight, including at the top of the path.
  8. 8
    commons.wikimedia.org/wiki/File:Billy_Wallace_kicking.jpg
    • Photo: Billy Wallace kicking for conversion during the Original All Blacks 1905 tour (from The Complete Rugby Footballer, 1906).

Image and sound credits

  • Billy Wallace kicking for goal, 1905 (The Complete Rugby Footballer, 1906, D. Gallaher & W. Stead): public domain, via Wikimedia Commons
  • Projectile trajectories, Niccolò Tartaglia, Nova scientia (1537): public domain, via Wikimedia Commons (scan: Max Planck Institute for the History of Science, ECHO)
  • Portrait of Galileo Galilei, Justus Sustermans (1636), Royal Museums Greenwich: public domain, via Wikimedia Commons
  • Title page, Discorsi e dimostrazioni matematiche intorno a due nuove scienze (1638), Posner Library, Carnegie Mellon University: public domain, via Wikimedia Commons
  • Pages 241–242, Galileo Galilei, Discorsi e dimostrazioni matematiche (Leiden, 1638), digitised by Google: public domain, via Wikimedia Commons
  • “ST vs CO 2012-03-10 – Conversion kick by Luke McAlister” by Pierre-Selim, CC BY 3.0 (https://creativecommons.org/licenses/by/3.0), via Wikimedia Commons
  • Diagrams and trajectories drawn in code from the equations of motion (g = 9.8 m/s², no air resistance)
  • Paper textures: ambientCG (CC0) · Torn paper edges generated with AI
  • Sound effects: Freesound contributors (CC0)
  • Music: generated with ACE-Step · Voice: Gemini TTS

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