Special Cases of Particle Motion

Many kinematics problems reduce to several standard cases of particle motion. They are conveniently classified by trajectory shape and by how velocity changes.

Uniform rectilinear motion

If a particle moves along a straight line with constant algebraic velocity $v$, its coordinate follows $s=s_0+vt$. Acceleration is zero.

Uniformly accelerated rectilinear motion

For rectilinear motion with constant algebraic acceleration $a$:

$$v=v_0+at,$$

$$s=s_0+v_0t+\frac{at^2}{2},$$

$$v^2=v_0^2+2a(s-s_0).$$

The signs of $v_0$ and $a$ specify the initial direction of motion and the acceleration direction relative to the selected axis.

If $a$ and $v_0$ have the same sign, speed initially increases. If their signs are opposite, the particle may slow to rest and then reverse direction.

Free fall as a special case

If air resistance and variation of gravitational acceleration with altitude are neglected, vertical motion near Earth’s surface is uniformly accelerated with $\vec g$ directed downward. Signs in the scalar equations depend on the selected positive vertical direction.

Uniform circular motion

For constant speed $v$ on a circle of radius $R$, tangential acceleration is zero but normal acceleration is not: $a_n=v^2/R$. It points toward the circle center, so the velocity vector continuously changes direction.

Nonuniform circular motion

If speed changes, the particle has both tangential acceleration $a_\tau=dv/dt$ and normal acceleration $a_n=v^2/R$. Total acceleration is their vector sum.

Example

A car moves along a straight line with $v_0=5$ m/s and constant acceleration $a=2$ m/s². After 4 s, its velocity is $v=5+2\cdot4=13$ m/s and its displacement from the initial position is $5\cdot4+2\cdot4^2/2=36$ m.

Common mistakes

• using constant-acceleration formulas when $a$ varies with time;
• assuming acceleration is zero in uniform circular motion;
• substituting $g$ without matching its sign to the chosen axis direction;
• confusing coordinate $s$ with distance traveled when the particle reverses direction.