Elementary motions

Two of the most common motions are straight-line and circular kinematics. While the general kinematic formulas apply to these situations, we can write simpler formulas just for these special cases.

Rectilinear motion

Rectilinear motion is motion in a straight line. In the particular case that acceleration is constant, the velocity and position can be found by explicit integration.

Constant linear acceleration.

\[\begin{aligned} a &= a_0 = \text{constant} \\ v &= v_0 + a_0 t \\ x &= x_0 + v_0 t + \frac{1}{2} a_0 t^2 \end{aligned}\]

If the $x$ and $y$ components of a 2D motion are independent, then each direction can be regarded as simple rectilinear motion. For example, if a projectile flies through the air then the horizontal component is linear in $t$, while the vertical motion with constant gravitational acceleration is a parabola.

Circular motion

Motion in a circle of constant radius gives velocity in the tangential direction $\hat{e}_\theta$. The acceleration has one component directly inwards in the radial direction (centripetal acceleration) and one tangential component due to angular acceleration $\alpha$.

Circular motion (constant $r$).

\[\begin{aligned} r &= \text{constant} \\ \vec{v} &= r \omega \,\hat{e}_\theta \\ \vec{a} &= - r \omega^2 \,\hat{e}_r + r \alpha \,\hat{e}_\theta \end{aligned}\]

Note that for circular motion the velocity $v$ is linear in both the radius $r$ and angular velocity $\omega$. The tangential acceleration is linear in both the radius $r$ and angular acceleration $\alpha$, while the radial acceleration is linear in $r$ but quadratic in $\omega$.

In the case of constant angular acceleration, the angular components function like rectilinear motion but in a circle, giving the explicit formulas:

Constant angular acceleration.

\[\begin{aligned} r &= r_0 = \text{constant} \\ \alpha &= \alpha_0 = \text{constant} \\ \omega &= \omega_0 + \alpha_0 t \\ \theta &= \theta_0 + \omega_0 t + \frac{1}{2} \alpha_0 t^2 \end{aligned}\]