Projectile Motion

Anything you throw follows a parabola. Its height is a quadratic in time, so the quadratic formula tells you exactly when it lands.

range 27.28 mapex 9.66 mflight 2.65 s

How it works

Quadratic functions

In one sentence

Without air resistance, a projectile launched at speed $v_0$ and angle $\theta$ from height $h_0$ has height $h(t) = -\tfrac12gt^2 + v_0\sin\theta\,t + h_0$ and horizontal distance $x(t) = v_0\cos\theta\,t$ .

General formula

{\htmlData{sym=0}{h(t)}} = -\tfrac12 {\htmlData{sym=1}{g}} {\htmlData{sym=5}{t}}^2 + {\htmlData{sym=2}{v_0}}\sin{\htmlData{sym=3}{\theta}}\,{\htmlData{sym=5}{t}} + {\htmlData{sym=4}{h_0}}

Gravity only pulls down, so height is a quadratic in time while sideways motion keeps a constant speed; together they trace a parabola.

What the symbols mean

$h(t)$
Height
How high the ball is t seconds after launch.
$g$
Gravity
Pulls everything down at 9.8 m/s² near Earth's surface.
$v_0$
Launch speed
How fast the ball leaves the launcher.
$\theta$
Launch angle
The angle above horizontal the ball is fired at.
$h_0$
Launch height
The height the ball starts from.
$t$
Time
Seconds since launch.

Gravity supplies the $-\tfrac12 g t^2$ term. Launch speed and angle set the linear term.
The vertex, at $t = \dfrac{v_0\sin\theta}{g}$ , is the highest point.
Setting $h(t) = 0$ and applying the quadratic formula gives the landing time.
Horizontal distance grows linearly: $x(t) = v_0\cos\theta\,t$ .

Learn the math

PolynomialsAlgebra · Lesson 4, Preliminaries Quadratic Equations by Factoring and Square RootsAlgebra · Lesson 11, Solving Equations and Inequalities Completing the Square and the Quadratic FormulaAlgebra · Lesson 12, Solving Equations and Inequalities Applications of Quadratic EquationsAlgebra · Lesson 13, Solving Equations and Inequalities Graphing and InterceptsAlgebra · Lesson 21, Graphing and Functions ParabolasAlgebra · Lesson 28, Common Graphs Parametric Equations and CurvesCalc II · Lesson 16, Parametric Equations and Polar Coordinates Velocity and AccelerationCalc III · Lesson 15, Three-Dimensional Space

Learn the math

PolynomialsAlgebra · Lesson 4, Preliminaries

Quadratic Equations by Factoring and Square RootsAlgebra · Lesson 11, Solving Equations and Inequalities

Completing the Square and the Quadratic FormulaAlgebra · Lesson 12, Solving Equations and Inequalities

Applications of Quadratic EquationsAlgebra · Lesson 13, Solving Equations and Inequalities

Graphing and InterceptsAlgebra · Lesson 21, Graphing and Functions

ParabolasAlgebra · Lesson 28, Common Graphs

Parametric Equations and CurvesCalc II · Lesson 16, Parametric Equations and Polar Coordinates

Velocity and AccelerationCalc III · Lesson 15, Three-Dimensional Space