1D and 2D Kinematics

Kinematics describes motion—position, velocity, and acceleration as functions of time—without asking what forces cause that motion. From here on, calculus is assumed: instantaneous rates are derivatives of position or velocity, and changes over an interval can be recovered with integrals when you know how acceleration varies.

Useful Variables

$t$ = time (Units: seconds ($s$))
$x$ or $s$ = displacement (Units: meters ($m$))
$v$ = velocity (Units $\frac{m}{s}$)
$a$ = acceleration (Units $\frac{m}{s^2}$)
$g$ = acceleration due to Earth’s gravity = $9.8 \frac{m}{s^2}$
$h$ = height (Units: $m$)
$R$ = range (Units: $m$)

Scalars, vectors, and describing motion

A scalar has magnitude only (examples: speed, distance, time). A vector has magnitude and direction (examples: displacement, velocity, acceleration). In one dimension, a sign attached to a scalar component encodes direction along an axis.

Choose an origin and a positive direction along each axis. Displacement $\Delta x$ is the change in position; distance is the length of the path traveled and is always nonnegative. Average velocity over an interval is

\[\bar{v} = \frac{\Delta x}{\Delta t}.\]

Instantaneous velocity is the time derivative of position when position is given as a function $x(t)$:

\[v = \frac{dx}{dt}.\]

Speed is the magnitude of velocity, $$

$$, in one dimension.

Average acceleration is

\[\bar{a} = \frac{\Delta v}{\Delta t}.\]

Instantaneous acceleration is the derivative of velocity with respect to time, equivalently the second derivative of position:

\[a = \frac{dv}{dt} = \frac{d^2x}{dt^2}.\]

If you know $a(t)$, the change in velocity between times $t_1$ and $t_2$ follows from integration:

\[v(t_2) - v(t_1) = \int_{t_1}^{t_2} a(t)\, dt,\]

and similarly position from velocity.

Constant acceleration in one dimension

Many problems use constant acceleration $a$ (free fall near Earth’s surface is a common case with $a = -g$ or $a = +g$ depending on axis choice). With initial position $x_0$ and initial velocity $v_0$ at $t = 0$,

$\Delta x = vt$ (if $a$ = 0)
\[\Delta v = at\]
\[\Delta x = v_0 \Delta t + \frac{1}{2} a (\Delta t)^2\]
\[\Delta x = \Delta v_f t - \frac{1}{2} a (\Delta t)^2\]
\[v^2 = v_0^2 + 2a(\Delta x)\]

These are algebraic consequences of $a = dv/dt$ constant and $v = dx/dt$. The five equations are sometimes called the Big Five.

is useful when time is unknown but initial and final velocities are known. Always check that your signs for $v_0$ and $a$ match the coordinate system.

Two dimensions and projectile motion

In 2D, vectors can be broken down into $x$- and $y$-components. For projectile motion with negligible air resistance, horizontal acceleration is zero and vertical acceleration is $g$ downward (again, signs depend on whether you call “up” positive $y$ or not). The motions along $x$ and $y$ are independent except that they share the same time parameter $t$.

With initial speed $v_0$ at launch angle $\theta$ above the horizontal,

\[v_{0x} = v_0 \cos\theta, v_{0y} = v_0 \sin\theta.\]

Typical component equations (up is positive $y$, gravity is downward) then become:

\[x = x_0 + v_{0x} t, \qquad y = y_0 + v_{0y} t - \frac{1}{2} g t^2,\] \[v_x = v_{0x}, \qquad v_y = v_{0y} - gt.\]

Remember: ALWAYS make sure you know which direction you define as positive y! The trajectory in the vertical plane is a parabola until the object hits something.

If launch and landing occur at the same height, the shortcuts for range, height, and time on level ground is:

$R = \frac{v_0^2 \sin(2\theta)}{g},$,

$h = \frac{v_0^2 \sin^2 (\theta)}{2g},$, and

\[t = \frac{v_0 \sin(\theta)}{g},\]

If launch and landing heights differ, solve the quadratic in $t$ from the $y$ equation rather than memorizing these shortcuts.

Relative velocity (introduction)

The velocity of object $A$ relative to object $B$ is written $\vec{v}_{A/B}$. With three objects (or frames), the usual composition rule is

\[\vec{v}_{A/C} = \vec{v}_{A/B} + \vec{v}_{B/C}.\]

Add vectors component-wise. This idea appears again with rotating frames later; for now, restrict to inertial frames in uniform relative motion. You can find more on the USAPhO section on mechanics.