Kinematics of a particle

Marcos Duarte

Kinematics is the branch of classical mechanics that describes the motion of objects without consideration of the causes of motion (Wikipedia). Kinematics of a particle is the description of the motion when the object is considered a particle. A particle as a physical object does not exist in nature; it is a simplification to understand the motion of a body or it is a conceptual definition such as the center of mass of a system of objects.

Position

Consider a point in the three-dimensional Euclidean space described in a Cartesian coordinate system:

Figure. Representation of a point **P** and its position vector **a** in a Cartesian coordinate system. The versors i, j, k form a basis for this coordinate system and are usually represented in the color sequence RGB (red, green, blue) for easier visualization.

The position of this point in space can be represented as a triple of values each representing the coordinate at each axis of the Cartesian coordinate system following the $ \mathbf{X, Y, Z} $ convention order (which is omitted):

$$ (x,\: y,\: z) $$

The position of a particle in space can also be represented by a vector in the Cartesian coordinate system, with the origin of the vector at the origin of the coordinate system and the tip of the vector at the point position:

$$ \mathbf{r}(t) = x\:\mathbf{i} + y\:\mathbf{j} + z\:\mathbf{k} $$

Where $ \mathbf{i,\: j,\: k} $ are unit vectors in the directions of the axes $ \mathbf{X, Y, Z} $.

For a review on vectors, see the notebook Scalar and vector.

With this new notation, the coordinates of a point representing the position of a particle that vary with time would be expressed by the following position vector $ \mathbf{r}(t)$:

$$ \mathbf{r}(t) = x(t)\:\mathbf{i} + y(t)\:\mathbf{j} + z(t)\:\mathbf{k} $$

A vector can also be represented in matrix form:

$$ \mathbf{r}(t) = \begin{bmatrix} x(t) \\y(t) \\z(t) \end{bmatrix}$$

And the unit vectors in each Cartesian coordinate in matrix form are given by:

$$ \mathbf{i}= \begin{bmatrix}1\\0\\0 \end{bmatrix},\; \mathbf{j}=\begin{bmatrix}0\\1\\0 \end{bmatrix},\; \mathbf{k}=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$$

In linear algebra, a set of unit linearly independent vectors as the three vectors above (orthogonal in the Euclidean space) that can represent any vector via linear combination is called a basis. A basis is the foundation of creating a reference frame and we will study how to do that other time.

Displacement

The shortest distance from the initial to the final position of a particle. As the difference between two vectors; displacement is also a vector quantity.

Velocity

Velocity is the rate (with respect to time) of change of the position of a particle:

$$ \mathbf{v}(t) = \frac{\mathbf{r}(t_2)-\mathbf{r}(t_1)}{t_2-t_1} = \frac{\Delta \mathbf{r}}{\Delta t}$$

The instantaneous velocity of the particle is obtained when $\Delta t\;$ approaches to zero, which from calculus is the first-order derivative of the position vector. The derivative of a vector is obtained by differentiating each vector component:

$$ \mathbf{v}(t) = \frac{\mathrm{d}\mathbf{r}(t)}{dt} = \frac{\mathrm{d}x(t)}{\mathrm{d}t}\mathbf{i} + \frac{\mathrm{d}y(t)}{\mathrm{d}t}\mathbf{j} + \frac{\mathrm{d}z(t)}{\mathrm{d}t}\mathbf{k} $$

And in matrix form:

$$ \mathbf{v}(t) = \begin{bmatrix} \frac{\mathrm{d}x(t)}{\mathrm{d}t} \\ \frac{\mathrm{d}y(t)}{\mathrm{d}t} \\ \frac{\mathrm{d}z(t)}{\mathrm{d}t} \end{bmatrix}$$

Acceleration

Acceleration is the rate (with respect to time) of change of the velocity of a particle, which can also be given by the second-order rate of change of the position:

$$ \mathbf{a}(t) = \frac{\mathbf{v}(t_2)-\mathbf{v}(t_1)}{t_2-t_1} = \frac{\Delta \mathbf{v}}{\Delta t} = \frac{\Delta^2 \mathbf{r}}{\Delta t^2}$$

Likewise, acceleration is the first-order derivative of the velocity or the second-order derivative of the position vector:

$$ \mathbf{a}(t) = \frac{\mathrm{d}\mathbf{v}(t)}{\mathrm{d}t} = \frac{\mathrm{d}^2\mathbf{r}(t)}{\mathrm{d}t^2} = \frac{\mathrm{d}^2x(t)}{\mathrm{d}t^2}\mathbf{i} + \frac{\mathrm{d}^2y(t)}{\mathrm{d}t^2}\mathbf{j} + \frac{\mathrm{d}^2z(t)}{\mathrm{d}t^2}\mathbf{k} $$

And in matrix form:

$$ \mathbf{a}(t) = \begin{bmatrix} \frac{\mathrm{d}^2x(t)}{\mathrm{d}t^2} \\ \frac{\mathrm{d}^2y(t)}{\mathrm{d}t^2} \\ \frac{\mathrm{d}^2z(t)}{\mathrm{d}t^2} \end{bmatrix}$$

The antiderivative

As the acceleration is the derivative of the velocity which is the derivative of position, the inverse mathematical operation is the antiderivative (or integral):

$$ \begin{array}{l l} \mathbf{r}(t) = \mathbf{r}_0 + \int \mathbf{v}(t) \:\mathrm{d}t \\ \mathbf{v}(t) = \mathbf{v}_0 + \int \mathbf{a}(t) \:\mathrm{d}t \end{array} $$

Some cases of motion of a particle

Particle at rest

$$ \begin{array}{l l} \mathbf{a}(t) = 0 \\ \mathbf{v}(t) = 0 \\ \mathbf{r}(t) = \mathbf{r}_0 \end{array} $$

Particle at constant speed

$$ \begin{array}{l l} \mathbf{a}(t) = 0 \\ \mathbf{v}(t) = \mathbf{v}_0 \\ \mathbf{r}(t) = \mathbf{r}_0 + \mathbf{v}_0t \end{array} $$

Particle at constant acceleration

$$ \begin{array}{l l} \mathbf{a}(t) = \mathbf{a}_0 \\ \mathbf{v}(t) = \mathbf{v}_0 + \mathbf{a}_0t \\ \mathbf{r}(t) = \mathbf{r}_0 + \mathbf{v}_0t + \frac{\mathbf{a}_0t^2}{2} \end{array} $$

Visual representation of these cases

In [1]:
# Import the necessary libraries
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
import seaborn as sns
#sns.set()
sns.set_context("notebook", font_scale=1.2, rc={"lines.linewidth": 2, "lines.markersize": 10})
In [2]:
t = np.arange(0, 2.0, 0.02)
r0 = 1; v0 = 2; a0 = 4

plt.rc('axes',  labelsize=14,  titlesize=14) 
plt.rc('xtick', labelsize=10)
plt.rc('ytick', labelsize=10) 
f, axarr = plt.subplots(3, 3, sharex = True, sharey = True, figsize=(14,7))
plt.suptitle('Kinematics of a particle', fontsize=20);

tones = np.ones(np.size(t))

axarr[0, 0].set_title('at rest', fontsize=14);
axarr[0, 0].plot(t, r0*tones, 'g', linewidth=4, label='$r(t)=1$')
axarr[1, 0].plot(t,  0*tones, 'b', linewidth=4, label='$v(t)=0$')
axarr[2, 0].plot(t,  0*tones, 'r', linewidth=4, label='$a(t)=0$')
axarr[0, 0].set_ylabel('r(t) [m]')
axarr[1, 0].set_ylabel('v(t) [m/s]')
axarr[2, 0].set_ylabel('a(t) [m/s$^2$]')

axarr[0, 1].set_title('at constant speed');
axarr[0, 1].plot(t, r0*tones+v0*t, 'g', linewidth=4, label='$r(t)=1+2t$')
axarr[1, 1].plot(t, v0*tones,      'b', linewidth=4, label='$v(t)=2$')
axarr[2, 1].plot(t,  0*tones,      'r', linewidth=4, label='$a(t)=0$')

axarr[0, 2].set_title('at constant acceleration');
axarr[0, 2].plot(t, r0*tones+v0*t+1/2.*a0*t**2,'g', linewidth=4,
                 label='$r(t)=1+2t+\\frac{1}{2}4t^2$')
axarr[1, 2].plot(t, v0*tones+a0*t,             'b', linewidth=4,
                 label='$v(t)=2+4t$')
axarr[2, 2].plot(t, a0*tones,                  'r', linewidth=4,
                 label='$a(t)=4$')

for i in range(3): 
    axarr[2, i].set_xlabel('Time [s]');
    for j in range(3):
        axarr[i,j].set_ylim((-.2, 10))
        axarr[i,j].legend(loc = 'upper left', frameon=True, framealpha = 0.9, fontsize=16)
        
plt.subplots_adjust(hspace=0.09, wspace=0.07)

Kinematics of human movement

An example where the analysis of some aspects of the human body movement can be reduced to the analysis of a particle is the study of the biomechanics of the 100-metre race. Watch the video below to understand how this can be done.

In [1]:
from IPython.display import IFrame
IFrame('http://video.mit.edu/embed/12042/', width=640, height=360)
Out[1]:

A technical report with the kinematic data for the 100-m world record by Usain Bolt discussed in the video above can be downloaded from the website for Research Projects from the International Association of Athletics Federations. Here is a direct link for that report. In particular, the following table shows the data for the three medalists in that race:

partial times of the 100m-race at Berlin 2009
Figure. Data from the three medalists of the 100-m dash in Berlin, 2009 (IAAF report).

The column RT in the table above refers to the reaction time of each athlete. The IAAF has a very strict rule about reaction time: any athlete with a reaction time less than 100 ms is disqualified from the competition! See the website Reaction Times and Sprint False Starts for a discussion about this rule.

You can measure your own reaction time visiting this website: http://www.humanbenchmark.com/tests/reactiontime.

The article A Kinematics Analysis Of Three Best 100 M Performances Ever by Krzysztof and Mero presents a detailed kinematic analysis of 100-m races.

Problems

  1. Consider the data for the three medalists of the 100-m dash in Berlin, 2009, shown previously.
    a. Calculate the instantaneous velocity and acceleration.
    b. Plot the graphs for the displacement, velocity, and acceleration versus time.
    c. Plot the graphs velocity and acceleration versus partial distance (every 20m).
    d. Calculate the mean velocity and mean acceleration and the instants and values of the peak velocity and peak acceleration.

  2. The article "Biomechanical Analysis of the Sprint and Hurdles Events at the 2009 IAAF World Championships in Athletics" by Graubner and Nixdorf lists the 10-m split times for the three medalists of the 100-m dash in Berlin, 2009:

    partial times of the 100m-race at Berlin 2009

    a. Repeat the same calculations performed in problem 1 and compare the results.

  3. A body attached to a spring has its position (in cm) described by the equation $x(t) = 2\sin(4\pi t + \pi/4)$.
    a. Calculate the equation for the body velocity and acceleration.
    b. Plot the position, velocity, and acceleration in the interval [0, 1] s.

  4. There are some nice free software that can be used for the kinematic analysis of human motion. Some of them are: Kinovea, Tracker, and SkillSpector. Visit their websites and explore these software to understand in which biomechanical applications they could be used.

References