Rectilinear Motion

Rectilinear motion describes a particle travelling along a straight line. A single coordinate $s$, measured from a fixed origin, fully describes its position.

It can be considered the simplest case of curvilinear motion, where the radius of curvature $\rho \to \infty$. Motion along each axis of rectangular coordinates is treated as rectilinear.

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Position, Velocity and Acceleration

Motion is measured along the line with a sign convention where a given direction is defined as positive.

Quantity	Definition	Meaning
Position	$s$	Location relative to origin
Velocity	$v=\frac{ds}{dt}=\dot{s}$	Rate of change of position
Acceleration	$a=\frac{dv}{dt}=\frac{d^{2}s}{d{t}^2}=\ddot{s}$	Rate of change of velocity

Displacement vs Distance

Displacement ($\Delta s$) is the change in position of an object. It can be positive or negative depending on the direction of displacement.

Distance travelled ($d$) is the total path length and is always positive. If the path reverses direction, each segment of displacement (separated by $v=0$) must be added together to find total distance.

Quote bq

An object speeds up when $v$ and $a$ are in the same direction, and slows down when they are in different directions.

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Eliminating Time

Dividing $a=\frac{dv}{dt}$ by $v=\frac{ds}{dt}$ removes $t$ and gives a third relation:
$vdv=ads$

The form to integrate depends on how acceleration is given:

Given	Relation to use	Integrate
$a=f(t)$	$dv=adt,ds=vdt$	$\int dv=\int adt$, then $\int ds=\int vdt$
$a=f(v)$	$dt=\frac{dv}{a}$ or $ds=\frac{vdv}{a}$​	separate variables $v$ and $t$ (or $s$)
$a=f(s)$	$vdv=ads$	$\int vdv=\int ads$

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SUVAT Equations

When acceleration is constant, integrating gives the following equations:
$v=u+at$
$s=ut+\frac{1}{2}a{t}^2$
$v^2=u^2+2as$
$s=\frac{v+u}{2}t$
$s=vt-\frac{1}{2}a{t}^2$

Free Fall

Motion under gravity has constant acceleration where:
$a_y=-g=-9.81{\text{ms}}^{-2}$

Projectile Motion

Projectile motion is the motion of a particle moving under gravity alone. It is a special case of curvilinear motion, where acceleration is constant and vertical.

Horizontal and vertical motion are independent. Each axis is its own rectilinear motion problem sharing only time $t$.

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Acceleration as a Function of Position

When $a$ depends on $s$, use $vdv=ads$. The most important case is when acceleration is proportional to displacement and opposite in sign:
$a=-k^{2}s⟹\frac{d^{2}s}{d{t}^2}+k^{2}s=0$

This has the general solution:
$s(t)=C_1\sin (kt)+C_2\cos (kt)$

• $C_1=\frac{v_0}{k}$ -> starting velocity
• $C_2=s_0$ -> starting displacement

Comment bq

This is known as simple harmonic motion and is the same equation governing free vibration.

Integral Identity

Position-dependent problems often integrate to an inverse sine:
$\int \frac{1}{\sqrt{A^2-(kx{)}^2}}dx=\frac{1}{k}\arcsin (\frac{kx}{A})$

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Related Motion

Relative Motion compares two particles on the same line. Dependent Motion links particles joined by cords, where differentiating the cord-length constraint gives the velocity relationship.

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