SOLUTION TO BASIC KINEMATICS PROBLEMS USING MATLAB® AND ITS TOOLBOXES Carlos A. Vanegas Universidad EAFIT, Medellín [email protected]

Abstract: This paper contains an approach to the solutions to three basic kinematics problems: one-dimension motion as a polynomial function of time, parabolic motion of projectiles, and circular motion on spherical bodies surfaces. Its purpose is to illustrate certain programming techniques in MATLAB® that provide analytical and simulated solutions to the given problems. Keywords: Forces, Friction, Function, Kinematics, Modeling, Motion, Programming, Simulation

1.

INTRODUCTION

Several kinematics problems are currently being studied through the use of different computational tools which provide analytical and simulated solutions. Many of these problems, such as those related to simple parabolic and circular motion, can be expressed and solved using classic mechanics equations, that after adequate calculations lead to exact solutions. Some other problems, like those that involve falling objects with air resistance, involve more complicated interactions between several forces, and thus, require the use of simulation which provides approximated solutions. In both cases, the use of certain computational tools is recommended. Not only does it make easier the necessary calculations, but also allows the construction of algorithms oriented to modeling and simulation. As an example of the mentioned fact, three basic kinematics problems have been selected to be solved

using such tools. All of them could be analytically solved just with elemental knowledge in kinematics and differential calculus, without the need of employing a computer. Nevertheless, it is shown herein how convenient could result developing a program on a specific computational language in order to enlarge the amount of input parameters for each problem, facilitate any adjustment of the values of the constants involved, and increase the options of data visualization. The computational tool used in this case was MATLAB®. MATLAB® is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation. (Mathworks Help). Some of the MATLAB features used in the solution of these three problems include matrix operations, data plotting, graphical user interface (GUI), and Symbolic Math Toolbox®.

Further solutions to more complex analyticallyunsolvable problems can be developed using Simulink®, a widely used software package for modeling and simulating dynamic systems.

Notice that the polynomial representing the position (Eq.1) is of degree 3, meaning that the position of the object varies proportionally to the cube of time (it also varies proportionally to the square of time). Therefore, the polynomial representing the first order derivative ( v x ) is of degree 2, and the one representing the second order derivative ( a x ) is

USING MATHEMATICAL OPERATIONS TO OBTAIN MOTION EQUATIONS AND DATA TO BE PLOTTED (PROBLEM 1)

linear. This means that the object’s velocity, given by equation 4, varies proportionally to the square of time.

2.1. Position of an object in a one-dimension motion as a polynomial function of time

2.2. Understanding the object’s motion based on plotted data.

The first problem deals with the varying position of an object described by a continuous function of time:

Up to now, the equations required to find the position, velocity, and acceleration of the body for any given time have been obtained, and substituting any positive value of t into equations (1), (4), and (5), would return the value of x , v x , and a x

An object moves along the X-axis in accordance with the following law:

x(t ) = 2t 3 + 5t 2 + 5

(1)

a) Find the position, velocity, and acceleration of the body for any given time. b) Plot the acceleration as a function of time for any interval entered by the user It is clear that the first part of the problem can be analytically solved by referring to the definition of instantaneous velocity and acceleration as the first and second order derivatives of the position function with respect to time (Serway, 2000). That is:

∆x dx = ∆t →0 ∆t dt

v x = lim

∆v x dv x d  dx  d 2 x = =  = 2 ∆t →0 ∆t dt dt  dt  dt

a x = lim

respectively. In addition to this, an idea of the motion of the object could be formed based on these equations. As an example of this, it can be seen how the relation between the object’s acceleration and time, given by equation 5, is linear. This leads to foresee that a plot of a x vs. t, would result in a straight line with a slope of 12, that crosses the a x axis at 12.

Acceleration vs Time 60

50

(2)

(3)

The following functions are obtained when solving the first and second order derivatives for x(t ) .

Acceleration (ft/s2)

2.

40

30

20

10

0

v x = 6t + 10t a x = 12t + 10 2

(4) (5)

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

4

Fig. 1. Variation of the object’s acceleration as a function of time between t=0, and t=4.

A recommended way to solve this problem using MATLAB® is as follows: Define t as a symbolic variable using the command ‘sym’; Store equation (1) in a variable x, as function of the symbolic variable t; Use the function ‘diff’ from the Symbolic Math Toolbox® to obtain the first and second order derivatives; Use ‘eval’ or ‘feval’ to evaluate the obtained functions for any given value of t. To plot the acceleration as a function of time, it is necessary to define a vector T containing a finite number of positive values for t, and then for this vector, evaluate the function obtained for the acceleration, and store it in a variable A. After this, plot both vectors (A vs. T) using the command ‘plot’. Since the values of t and the lower and upper limits of the interval in which data will be plotted are entered by the user, it is necessary to provide convenient and simple data-input-objects. Graphic User Interfaces, are recommended for this purpose.

3.

VALIDATING DATA INPUT AND USING MULTIPLE PLOTS TO ANALYZE PARABOLIC MOTION OF PROJECTILES

3.1. Data validation based on parabolic motion equations. The second problem deals with the description of the trajectory followed by an object under a parabolic motion. A projectile is thrown with an initial velocity vi and an angle

θ

in relation to the horizontal.

a) Find the object’s position and velocity for a given time t . b) Obtain for different time intervals the following six plots: v x vs. t , v y vs. t , x vs.

t , y vs. t , x vs. y , a vs. t . A projectile is an object which is thrown with some initial velocity, and then allowed to be acted upon by the forces of gravity and possible drag. The maximum upward distance h reached by the

projectile is called the height, the horizontal distance traveled is called the range, and the path of the object is called its trajectory (O’Hanian, 1989). If no drag forces are considered, the only force acting upon the projectile is gravity, which accelerates the object only in the downwards direction. Since no force is acting horizontally, the horizontal component of the velocity of the object remains constant. The angle at which the projectile is thrown in relation to the horizontal is denoted as θ , that is, the angle between the initial velocity vector, and the horizontal. According to these considerations, the following velocity equations can be obtained: v x = v xi = vi cos θ (6)

v y = v yi − gt = vi sin θ − gt

(7)

The following position equations are obtained by Integrating v x and v y with respect to time:

x = v xi t = (vi cos θ )t (8) 1 1 y = v yi t + a y t 2 = (vi sin θ )t − gt 2 (9) 2 2 Isolating t from equation 8, and substituting in equation 9, the equation of y as a function of x is obtained:

 g y = (tanθ ) x −  2 2  2vi cos θ

 2 x  

(10)

This expression has the form y = ax − bx , which is the equation of a parabola which crosses the origin. Is this the reason why the trajectory of the object is considered parabolic. 2

It can be seen from equations 6, 7, 8 and 9, that to determinate the position and velocity of the projectile for any given time t, it is necessary and sufficient to know the values of the constants vi and θ . Thus, in this case, the user must be asked to enter these two constants. However, only numeric values in certain intervals are valid to be substituted in any of the equations. Since this model represents a real-world situation, the mentioned intervals must be determined

so that the possible values remain coherent with the situation. As an example of this, the interval for which values of θ are allowed can be found taking into account the following: 1) Since the object is considered to be initially laying on a surface, this angle should not be smaller than 0º because it would mean that the object would collide against the ground; 2) Since the equations were built considering a motion of the object along the positive x-axis, θ should not be greater than 90º because this would imply a backwards motion. For analog reasons, vi must be a positive value. A process used to determine if data are accurate, complete, or meet specified criteria is known as Data Validation (ISO/IEC, 1998). If using GUI in MATLAB®, this process can be done using a loop such as ‘while’ function. This function repeats statements an indefinite number of times while a certain condition is true.

Therefore, a straight line with slope g ≈ −9.8 that intercepts the v y -axis at vi sin θ is expected to be the plot of the second relation ( v y vs. t ). For the next two relations: x vs. t , y vs. t , it can be concluded based on equations 8 and 9 that plotting the relation between x and time results in a straight line with slope vi cos θ that crosses the origin, and that plotting y vs. t , results in an openingdownwards parabola that crosses the time-axis at t=0 and t=’Time of Flight’, with vertex on ( 0.5t ,‘maximum upward distance’). Y vs. Time 3

2.5

2

3.2. Analyzing parabolic motion based on multiple data plots.

Y (m)

1.5

1

0.5

For this problem, six different relations between variables are recommended to be plotted for intervals entered by the user. Two of them are v x vs. t , and

v y vs. t . Since the horizontal velocity of the object remains constant throughout time, it is expected to obtain a straight horizontal line in the first plot. The vertical velocity of the object is maximum for t=0, and starts decreasing at a constant rate g. V y vs . t 8 6 4

V y (m / s )

2 0 -2 -4 -6 -8

0

0.5

1 Tim e (s )

Fig. 2. Variation of the projectile’s vertical velocity as function of time. (0 < t < ‘Time of Flight’).

1.5

0

-0.5

0

0.5

1

1.5

Time (s)

Fig. 3. Variation of projectile’s height as a function of time. (0 < t < ‘Time of Flight’). The last two plots involve the relation between the projectile’s acceleration and time, and the relation between the height and the x position of the object. The first one would result again in an horizontal straight line a x ≈ −9.8 . The second one corresponds to the object’s trajectory; that is an opening-downwards parabola that crosses the x-axis at x=0 and x= ‘range’, with vertex on ( 0.5 * ‘range’, ‘maximum upward distance’). Using the calculus definition of the maximum value of a function, the ‘maximum upward distance’ and ‘range’ values can be obtained. The range is defined as the horizontal distance traveled by the object during the time of flight. Notice at this point that data validation should again be considered to avoid the user entering values of time greater than the time of flight, or smaller than zero.

4.

ADITIONAL PROBLEM - ACCELERATION AND VELOCITY OF A POINT ON A ROTATING SOLAR SYSTEM BODY’S SURFACE -

The earth rotates at a rate of ϖ = 7.292 × 10 s Find as a function of latitude, the acceleration and velocity of a point on its surface. Plot the relations between velocity and time, and acceleration and time. −5

−1

minimum values at λ = ±90º (the Poles). Thus, from equations 13 and 14 it can be seen that vt and

ar vary proportionally to cos λ , and therefore it can be concluded that the maximum values for velocity and acceleration are those presented by the points located on the Equator. (Figures 4 and 5 illustrate this fact). L a t it u d e vs . Ta n g e n t ia l V e lo c it y 500

450

400

Ta n g e n t ia l V e lo c it y (m / s )

350

The third problem deals with the relation existing between latitude, velocity and acceleration for a point located on the surface of a rotating sphere. The expression for the linear velocity of the point is:

300

250

200

150

100

vt = ϖ r

(11)

50

0 -1 0 0

r ⇒ r = Re cos λ Re

(12)

r = Re cos λ in equation 11, the linear velocity is

λ:

vt = ϖRe cos λ

20

40

60

80

100

Latitude vs . A c c eleration

0.03

0.025

0.02

0.015

0.005

0 -100

-80

-60

-40

-20

0 Latitude (deg)

20

40

60

80

100

(13)

2

vt = ϖ 2 r = ϖ 2 Re cos λ r

0 L a t it u d e (d e g )

Fig. 5. Variation of Acceleration as a function of Latitude λ . (-90º< λ < 90º ).

The expression that describes the acceleration in a circular motion is:

ar =

-2 0

0.01

where Re is the radius of the earth. Substituting obtained as a function of latitude

-4 0

0.035

2

cos λ =

-6 0

Fig. 4. Variation of Linear Velocity as a function of Latitude λ . (-90º< λ < 90º ).

A c c eleration (m /s )

where r is the perpendicular distance from the point to the axis of rotation. In the case of the earth, the perpendicular distance between a point on its surface and the axis of rotation (the line that goes from the geographic north pole to the geographic south pole) can be expressed as a function of latitude λ as follows (Heeke, 2003):

-8 0

(14)

Since the latitude λ varies from 90ºS to 90ºN, cos λ oscillates between 0 and 1, assuming a maximum value at λ = 0º (the Equator line), and

Other solid solar system bodies in which there is particular interest (such as the Moon and Mars) perform a circular motion similar to the Earth’s. Obtaining the values for the velocity and acceleration for a given point located on the surface of one of these bodies implies the knowledge of the constants ϖ and R p . To do so, if using a GUI, the user should be asked to select from a list the name of the body, and a code that assigns values to these constants depending on the item selected should be generated.

CONCLUSIONS

REFERENCES

The understanding of the motion of an object described by a position expression can be enhanced by obtaining the derivatives of first and second order of the expression with respect to time. Plotting these new functions that represent the object’s linear velocity and acceleration is recommended for such purpose. Certain MATLAB® features provide tools to facilitate these procedures. These tools can also be used to analyze additional problems concerning kinematics, such as the motion of projectiles describing a parabolic path, and the circular motion of points on a sphere’s surface. A way to perform such analysis involves changing the values of the constants to observe the behavior of the phenomena under different conditions. Particularly for these problems, the following observations were made using this technique: 1) The maximum range of a projectile following a parabolic path is achieved when the angle at which the projectile is thrown in relation to the horizontal is 45º; 2) The maximum height achieved by an object following a parabolic path is achieved when the angle at which the projectile is thrown in relation to the horizontal is 90º; 3) The maximum values for velocity and acceleration of a point located on a space body’s surface are those presented by the points located on the body’s Equator. 4) The linear velocity and acceleration of a point located on the Earth’s surface at a latitude λ is greater than the ones of a point located at either Mars or the Moon’s surface at the same latitude. 5) The linear velocity and acceleration of a point located at the poles of a space body with respect the body’s axis of rotation is zero.

O'Hanian, H.C. "The Motion of Projectiles". In Physics Vol. 1, New York: W. W. Norton, pp. 71-76, 1985. Serway, R.A. and Beichner, R.J. “One-Dimension Motion”. In Physics for Scientists and Engineers, fifth English Edition. 2000 ISO/IEC 2382-8, Information Technology Vocabulary: Control, integrity, and security, 1998. Heeke, Franz. “Rotation of sun and planets”, http://www.surf2000.de/user/f-heeke/ article2.html . 2003 Meeus, Jean. “Astronomical formulae Calculators”. Fourth, Revised Edition. 1998

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