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Kinematics

  • Average vectors of velocity ($ \vec{v} $) and acceleration ($ \vec{a} $) of a point:
    $$\langle\vec{v} \rangle = \frac{\Delta \vec{r}}{\Delta t}, \langle\vec{a} \rangle = \frac{\Delta \vec{v}}{\Delta t},$$

    where $ \Delta\vec{r} $ is the displacement vector (an increment of a radius vector).

  • Velocity and acceleration of a point:
    $$\vec{v} = \frac{d \vec{r}}{d t}, \vec{a} = \frac{d \vec{v}}{d t}.$$
  • Acceleration of a point expressed in projections on the tangent and the normal to a trajectory:
    $$a_{\parallel} = \frac{d v_{\parallel}}{d t}, a_{\perp} = \frac{v^2}{R},$$

    where $ R $ is the radius of curvature of the trajectory at the given point.

  • Distance covered by a point:
    $$s = \int v dt,$$

    where $ v $ is the modulus of the velocity vector of a point.

  • Angular velocity and angular acceleration of a solid body:
    $$\vec{\omega} = \frac{d \vec{\phi}}{dt}, \vec{\alpha} = \frac{d\vec{\omega}}{dt}.$$
  • Relations between linear and angular quantities for a rotating solid body:
    $$ \vec{v} = \vec{\omega}\times \vec{r}, \omega_{\perp} = \omega^2 R, |\omega_\parallel| = \alpha,$$

    where $ \vec{r} $ is the radius vector of the considered point relative to an arbitrary point on the rotation axis.

 

Keywords: 
kinematics, vector, displacement, velocity, acceleration, angular velocity, angular acceleration
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