2D kinematics and projectile motion

Motion as a vector function

In two dimensions, position is a vector function of time:

ec{r}(t)=x(t)hat{i}+y(t)hat{j}

Velocity and acceleration are derivatives of this vector:

ec{v}(t)=rac{dec{r}}{dt}=rac{dx}{dt}hat{i}+rac{dy}{dt}hat{j}

ec{a}(t)=rac{dec{v}}{dt}=rac{d^2x}{dt^2}hat{i}+rac{d^2y}{dt^2}hat{j}

This means two-dimensional motion can often be treated as two linked one-dimensional motions: one in $x$ and one in $y$.

Projectile motion assumptions

Projectile motion describes an object launched through the air and moving under gravity. In the simplest model, air resistance is ignored and gravitational acceleration is constant and downward.

Choose $x$ horizontal and $y$ vertical, with upward positive. Then

$a_x=0$

$a_y=-g$

The horizontal velocity remains constant, while the vertical velocity changes due to gravity.

Initial velocity components

If a projectile is launched with initial speed $v_0$ at angle $heta$ above the horizontal, the components are

$v_{0x}=v_0cos heta$

$v_{0y}=v_0sin heta$

These components let us write separate equations for horizontal and vertical motion.

Horizontal motion:

$x(t)=x_0+v_{0x}t$

Vertical motion:

y(t)=y_0+v_{0y}t-rac{1}{2}gt^2

Vertical velocity:

$v_y(t)=v_{0y}-gt$

The path of a projectile

The projectile's path is a parabola in the ideal model. This can be shown by eliminating time. If $x_0=0$, then

t=rac{x}{v_0cos heta}

Substitute into the vertical equation:

y=x an heta-rac{g x^2}{2v_0^2cos^2 heta}

This is a quadratic function of $x$, so the trajectory is parabolic.

Time of flight and range

For a projectile that lands at the same vertical height from which it was launched, the time of flight is found by setting $y-y_0=0$:

0=v_0sin heta,t-rac{1}{2}gt^2

Besides $t=0$, the landing time is

T=rac{2v_0sin heta}{g}

The horizontal range is

$R=v_0cos heta,T$

Substituting gives

R=rac{v_0^2sin 2 heta}{g}

This range formula applies only when launch and landing heights are equal and air resistance is ignored.

Velocity direction changes

A projectile's velocity vector changes throughout the flight. The horizontal component remains constant, while the vertical component decreases linearly. At the top of the path, $v_y=0$, but $v_x$ is still nonzero. The projectile is still moving horizontally.

The acceleration remains downward at every point, including the top.

Independence of components

The powerful idea in projectile motion is independence of perpendicular components. Gravity affects vertical motion but not horizontal motion in the ideal model. This is why an object dropped from rest and an object launched horizontally from the same height hit the ground at the same time, if air resistance is negligible.

The big idea

Two-dimensional kinematics treats motion vectorially but solves it component by component. Projectile motion is the classic example: horizontal motion is constant velocity, vertical motion is constant acceleration. The result is a parabolic trajectory controlled by initial speed, launch angle, height, and gravity.