Kinematic quantities From the
instantaneous position , instantaneous meaning at an instant value of time , the instantaneous velocity and acceleration have the general, coordinate-independent definitions; \mathbf{v} = \frac{d \mathbf{r}}{d t} \,, \quad \mathbf{a} = \frac{d \mathbf{v}}{d t} = \frac{d^2 \mathbf{r}}{d t^2} Notice that velocity always points in the direction of motion, in other words for a curved path it is the
tangent vector. Loosely speaking, first order derivatives are related to tangents of curves. Still for curved paths, the acceleration is directed towards the
center of curvature of the path. Again, loosely speaking, second order derivatives are related to curvature. The rotational analogues are the "angular vector" (angle the particle rotates about some axis) , angular velocity , and angular acceleration : \boldsymbol{\theta} = \theta \hat{\mathbf{n}} \,,\quad \boldsymbol{\omega} = \frac{d \boldsymbol{\theta}}{d t} \,, \quad \boldsymbol{\alpha}= \frac{d \boldsymbol{\omega}}{d t} \,, where is a
unit vector in the direction of the axis of rotation, and is the angle the object turns through about the axis. The following relation holds for a point-like particle, orbiting about some axis with angular velocity : \mathbf{v} = \boldsymbol{\omega}\times \mathbf{r} where is the position vector of the particle (radial from the rotation axis) and the tangential velocity of the particle. For a rotating continuum
rigid body, these relations hold for each point in the rigid body.
Uniform acceleration The differential equation of motion for a particle of constant or uniform acceleration in a straight line is simple: the acceleration is constant, so the second derivative of the position of the object is constant. The results of this case are summarized below.
Constant translational acceleration in a straight line These equations apply to a particle moving linearly, in three dimensions in a straight line with constant
acceleration. Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one. \begin{align} v & = v_0 + a t & [1]\\ r & = r_0 + v_0 t + \tfrac12 {a}t^2 & [2]\\ r & = r_0 + \tfrac12 \left( v+v_0 \right )t & [3]\\ v^2 & = v_0^2 + 2a\left( r - r_0 \right) & [4]\\ r & = r_0 + vt - \tfrac12 {a}t^2 & [5]\\ \end{align} where: • is the particle's initial
position • is the particle's final position • is the particle's initial
velocity • is the particle's final velocity • is the particle's
acceleration • is the
time interval Equations [1] and [2] are from integrating the definitions of velocity and acceleration, In these variables, the equations of motion would be written \begin{align} v & = u + at & [1] \\ s & = ut + \tfrac12 at^2 & [2] \\ s & = \tfrac{1}{2}(u + v)t & [3] \\ v^2 & = u^2 + 2as & [4] \\ s & = vt - \tfrac12 at^2 & [5] \\ \end{align}
Constant linear acceleration in any direction The initial position, initial velocity, and acceleration vectors need not be collinear, and the equations of motion take an almost identical form. The only difference is that the square magnitudes of the velocities require the
dot product. The derivations are essentially the same as in the collinear case, \begin{align} \mathbf{v} & = \mathbf{a}t+\mathbf{v}_0 & [1]\\ \mathbf{r} & = \mathbf{r}_0 + \mathbf{v}_0 t + \tfrac12\mathbf{a}t^2 & [2]\\ \mathbf{r} & = \mathbf{r}_0 + \tfrac12 \left(\mathbf{v}+\mathbf{v}_0\right) t & [3]\\ \mathbf{v}^2 & = \mathbf{v}_0^2 + 2\mathbf{a}\cdot\left( \mathbf{r} - \mathbf{r}_0 \right) & [4]\\ \mathbf{r} & = \mathbf{r}_0 + \mathbf{v}t - \tfrac12\mathbf{a}t^2 & [5]\\ \end{align} although the
Torricelli equation [4] can be derived using the
distributive property of the dot product as follows: v^{2} = \mathbf{v}\cdot\mathbf{v} = (\mathbf{v}_0+\mathbf{a}t)\cdot(\mathbf{v}_0+\mathbf{a}t) = v_0^{2}+2t(\mathbf{a}\cdot\mathbf{v}_0)+a^{2}t^{2} (2\mathbf{a})\cdot(\mathbf{r}-\mathbf{r}_0) = (2\mathbf{a})\cdot\left(\mathbf{v}_0t+\tfrac{1}{2}\mathbf{a}t^{2}\right)=2t(\mathbf{a}\cdot\mathbf{v}_0)+a^{2}t^{2} = v^{2} - v_0^{2} \therefore v^{2} = v_0^{2} + 2(\mathbf{a}\cdot(\mathbf{r}-\mathbf{r}_0))
Applications Elementary and frequent examples in kinematics involve
projectiles, for example a ball thrown upwards into the air. Given initial velocity , one can calculate how high the ball will travel before it begins to fall. The acceleration is local acceleration of gravity . While these quantities appear to be
scalars, the direction of displacement, speed and acceleration is important. They could in fact be considered as unidirectional vectors. Choosing to measure up from the ground, the acceleration must be in fact , since the force of
gravity acts downwards and therefore also the acceleration on the ball due to it. At the highest point, the ball will be at rest: therefore . Using equation [4] in the set above, we have: s= \frac{v^2 - u^2}{-2g}. Substituting and cancelling minus signs gives: s = \frac{u^2}{2g}.
Constant circular acceleration The analogues of the above equations can be written for
rotation. Again these axial vectors must all be parallel to the axis of rotation, so only the magnitudes of the vectors are necessary, \begin{align} \omega & = \omega_0 + \alpha t \\ \theta &= \theta_0 + \omega_0t + \tfrac12\alpha t^2 \\ \theta & = \theta_0 + \tfrac12(\omega_0 + \omega)t \\ \omega^2 & = \omega_0^2 + 2\alpha(\theta - \theta_0) \\ \theta & = \theta_0 + \omega t - \tfrac12\alpha t^2 \\ \end{align} where is the constant
angular acceleration, is the
angular velocity, is the initial angular velocity, is the angle turned through (
angular displacement), is the initial angle, and is the time taken to rotate from the initial state to the final state.
General planar motion {{multiple image These are the kinematic equations for a particle traversing a path in a plane, described by position . They are simply the time derivatives of the position vector in plane
polar coordinates using the definitions of physical quantities above for angular velocity and angular acceleration . These are instantaneous quantities which change with time. The position of the particle is \mathbf{r} =\mathbf{r}\left ( r(t),\theta(t) \right ) = r \mathbf{\hat{e}}_r where and are the
polar unit vectors. Differentiating with respect to time gives the velocity \mathbf{v} = \mathbf{\hat{e}}_r \frac{d r}{dt} + r \omega \mathbf{\hat{e}}_\theta with radial component and an additional component due to the rotation. Differentiating with respect to time again obtains the acceleration \mathbf{a} =\left ( \frac{d^2 r}{dt^2} - r\omega^2\right )\mathbf{\hat{e}}_r + \left ( r \alpha + 2 \omega \frac{dr}{dt} \right )\mathbf{\hat{e}}_\theta which breaks into the radial acceleration ,
centripetal acceleration ,
Coriolis acceleration , and angular acceleration . Special cases of motion described by these equations are summarized qualitatively in the table below. Two have already been discussed above, in the cases that either the radial components or the angular components are zero, and the non-zero component of motion describes uniform acceleration.
General 3D motions In 3D space, the equations in spherical coordinates with corresponding unit vectors , and , the position, velocity, and acceleration generalize respectively to \begin{align} \mathbf{r} & =\mathbf{r}\left ( t \right ) = r \mathbf{\hat{e}}_r\\ \mathbf{v} & = v \mathbf{\hat{e}}_r + r\,\frac{d\theta}{dt}\mathbf{\hat{e}}_\theta + r\,\frac{d\varphi}{dt}\,\sin\theta \mathbf{\hat{e}}_\varphi \\ \mathbf{a} & = \left( a - r\left(\frac{d\theta}{dt}\right)^2 - r\left(\frac{d\varphi}{dt}\right)^2\sin^2\theta \right)\mathbf{\hat{e}}_r \\ & + \left( r \frac{d^2 \theta}{dt^2 } + 2v\frac{d\theta}{dt} - r\left(\frac{d\varphi}{dt}\right)^2\sin\theta\cos\theta \right) \mathbf{\hat{e}}_\theta \\ & + \left( r\frac{d^2 \varphi}{dt^2 }\,\sin\theta + 2v\,\frac{d\varphi}{dt}\,\sin\theta + 2 r\,\frac{d\theta}{dt}\,\frac{d\varphi}{dt}\,\cos\theta \right) \mathbf{\hat{e}}_\varphi \end{align} \,\! In the case of a constant this reduces to the planar equations above. ==Dynamic equations of motion==