Learning topic
Special Cases of Particle Motion
Uniform and uniformly accelerated rectilinear motion, circular motion and basic kinematic relations for a particle.
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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.