Predicting the Speed of a Gravity-Powered Vehicle - Report Example

PREDICTING THE SPEED OF A GRAVITY-POWERED VEHICLE A MATHEMATICAL MODEL Introduction Newton's second law of motion states that force is directly proportional to mass applied and the acceleration. It is denoted as. Gravity racers rely on gravitational forces to have forward motion. It is one of the simplest vehicle designs also known as soapboxes or downhill go-arts. It is a go-kart fitted with four-wheeled chassis. They are often raced during competitions down different runs for recreational purposes and time is recorded to define the performance criteria. With the absence of a motor, ease of construction and relatively low costs, gravity racing is a readily accessible sport to the public and ideal for the young people or organizations. The design of a gravity racer needs one to have a good grasp of some physics and engineering facts. This report seeks to understand how a gravity racer is built, how it operates, and its relationship with the newton's second law. Forces are involved heavily to ensure they provide a gravitational push. This paper will formulate a Newtonian model using the MATLAB software (version 7.10.0 r2010a). The gravity racer has been used to demonstrate the projectile trajectories in solving the newton's equation of linear motion and in the prediction of the speed of a track cyclist. A gravity racer moving downhill is subject to the forces such as those applied on a free-body diagram. Three main forces act in the direction of the motion of the vehicle. The forces are: (i) Force due to gravity (Fw) (ii) Air resistance during movement. Drag (FD) (iii) The frictional force between the vehicle and the surface. (FR) The frictional force is also referred to as the rolling resistance. The acceleration of the gravity racer can be obtained by resolving the forces acting upon it. The slope angle is observed by a gravity racer moving downhill. As a result, the force due to gravity is given by the equation below where m is the mass of the vehicle or racer, The air resistance force or the aerodynamic force, FD, acting against the moving racer is an attribute of the speed of the racer. V, the density of the air, ρ, and two design parameters namely the drag coefficient, CD, and the frontal area of the vehicle, A. The force is, therefore, obtained as, The gravity racer is a four-wheeled vehicle which encounters friction as it propels. The rolling resistance is a measure of the force needed to overcome friction between the racer's tires and the road surface. It varies depending on the average contact force. It is given by Replacing all the expressions of force in equation 1, Making acceleration, a, the subject Displacement is the measure moved by an object in a given direction. The forward Euler's method is used to obtain the velocity from acceleration and displacement from velocity for a given time-step (∆t). The solution can be obtained using the Euler method in which Where xn is the displacement at the time, t; and xn+1 is the displacement at time t+∆t. The information required to come up with a predictive model includes the mass of the soapbox, frontal area, and the race track's slope angle. The design of a gravity racer cart is formed of steel with four-wheels on the chassis. The designs can be simple, or even compound such as those made of carbon fiber and they feature a monocoque design. With the advent of innovation, there is some recent development in the design where electrical components have been added to the racer. It eliminates the idea of gravity being the single propeller. Methods The forward Euler’s method is used to find the velocity and displacement values of the gravity racer. (1) Flowchart The following flowchart shows how the program interacts to collect data and output different results on the figures and report findings. (2) Matlab code %% Gravity Racer % The general equation with all forces applied to a gravity racer % making the acceleration the subject of the equation % the equation is given as follows % a=gsin?-1/2m ?v^2 C_d A-g?_r cos? % using euler's method we obtain the velocity and displacement expressions % x_(n+1)=x_n+?t.v_n % v_(n+1)=v_n+?t.a_n clc; % clear command window clear all; % clear workspace close all; % close figUres format short; %% The standard coefficients and variables g = 9.80665; % acceleration of gravity (m/s^2) cd = 0.26; % drag coefficient (kg/s) m = 200; % mass of the gravity racer (kg) Ur = 0.004; % coefficient of rolling resistance A= 0.042; %frontal area A (m^2) thet=4; % slope angle (deg) theta=4*pi/180; % slope angle (rad) p=1.225; % density of the air (kg/m^3) v0 = 0; % initial velocity (m/s) convt=2.236936; % conversion from ms^-1 to mph % some calculated constants Ra=cd*A*p/(2*m); Rb =g*Ur*cos(theta); %% question 1 N=100; for i=1:N t(i)=i-1; v(i+1)= v0+g*sin(theta)*t(i); s(i+1)= 0.5*g*sin(theta)*t(i).^2; end figure(1) % val1 =abs(find(s==2000)); % v_at2000=v(val1) plot(s,v); grid on title('velocity vs. displacement (Fd=Fr=0)'); ylabel('velocity(m/s)'); xlabel('displacement (m)'); grid on % V(0)=0 for i=1:N t(i)=i-1; V(i+1)= (v0+g*sin(theta)*t(i))-Ra*v(i).^2.*t(i); %0.0000334425= ((cd*A*p)/(2*m))=0.0000334425 S(i+1)= g*sin(theta)*0.5*t(i).^2-Ra*v(i).^2.*0.5*t(i).^2; end figure(2) plot(S,V,'r'); grid on title('velocity vs. displacement (Fr=0, Fd~=0)'); ylabel('velocity(m/s)'); xlabel('displacement (m)'); axis([0 3500 0 70]) for i=1:N t(i)=i-1; V1(i+1)= (v0+g*sin(theta)*t(i))-(cd*A*p/(2*m)*v(i).^2.*t(i))-g*Ur*cos(theta)*t(i); S1(i+1)= g*sin(theta)*0.5*t(i).^2-cd*A*p/(2*m)*v(i).^2.*0.5*t(i).^2-0.5*g*Ur*cos(theta)*t(i).^2; end figure(3) plot(S1,V1,'g'); grid on title('Velocity vs. Displacement(Fr~=0, Fd~=0)'); ylabel('velocity(m/s)'); xlabel('displacement (m)'); axis([0 3500 0 70]) %% question 2 clc; clear all; disp('Choose a configuration for design') m=input('Enter the mass of the Racer:'); A=input('Enter a frontal Area:'); Cd=input('Enter the Drag Coefficient:'); Ur=input('Enter the coefficient of Rolling Resistance:'); %calling the racerTester function racerTester(m,A,Cd,Ur); function racerTester(m,A,Cd,Ur) %standard values g = 9.80665; % acceleration of gravity (m/s^2) rho=1.225; % density of the air (kg/m^3) v0 = 0; % initial speed (m/s) convt=2.236936; % conversion from ms^-1 to mph p=1.225; thet=4; % slope angle (deg) theta=4*pi/180; % slope angle (rad) % choosing the optimum combination %option 1: steel chassis and spoke wheels % m1=216-1; % Cd=0.31+0.01; % A1=0.5; % Ur1=0.003; % % % %option 2: steel chassis and disc wheels % m2=216+2; % A2=0.5; % Cd1=0.31-0.02; % Ur2=0.006; %option 3: carbon fiber monocoque and disc wheels % m2=162; % A2=0.36; % Cd1=0.20; % Ur2=0.006; %option 4: carbon fibre monocoque and spoke wheels % m2=159; % A2=0.36; % Cd1=0.23; % Ur2=0.003; % some calculated constants Ra=Cd*A*p/(2*m); Rb =g*Ur*cos(theta); N=100; for i=1:N t(i)=i-1; v(i+1)= v0+g*sin(theta)*t(i); s(i+1)= 0.5*g*sin(theta)*t(i).^2; end figure(1) % val1 =abs(find(s==2000)); % v_at2000=v(val1) plot(s,v); grid on title('velocity vs. displacement (Fd=Fr=0)'); ylabel('velocity(m/s)'); xlabel('displacement (m)'); grid on % V(0)=0 for i=1:N t(i)=i-1; V(i+1)= (v0+g*sin(theta)*t(i))-Ra*v(i).^2.*t(i); S(i+1)= g*sin(theta)*0.5*t(i).^2-Ra*v(i).^2.*0.5*t(i).^2; end figure(2) plot(S,V,'r'); grid on title('velocity vs. displacement (Fr=0, Fd~=0)'); ylabel('velocity(m/s)'); xlabel('displacement (m)'); axis([0 3500 0 70]) for i=1:N t(i)=i-1; V1(i+1)= (v0+g*sin(theta)*t(i))-(Ra*v(i).^2.*t(i))-g*Ur*cos(theta)*t(i); S1(i+1)= g*sin(theta)*0.5*t(i).^2-Ra*v(i).^2.*0.5*t(i).^2-0.5*g*Ur*cos(theta)*t(i).^2; end figure(3) plot(S1,V1,'g'); grid on title('Velocity vs. Displacement(Fr~=0, Fd~=0)'); ylabel('velocity(m/s)'); xlabel('displacement (m)'); axis([0 3500 0 70]) end %% question 3 clc; clear all; format short; thet=input('Enter The slope angle: '); %calling the function testSlope testSlope(thet); function testSlope(thet) %% The standard coefficients and variables g = 9.80665; % acceleration of gravity (m/s^2) cd = 0.26; % drag coefficient (kg/s) m = 200; % mass of the gravity racer (kg) Ur = 0.004; % coefficient of rolling resistance A= 0.042; %frontal area A (m^2) theta=thet*pi/180; % slope angle (rad) p=1.225; % density of the air (kg/m^3) v0 = 0; % initial velocity (m/s) convt=2.236936; % conversion from ms^-1 to mph % some calculated constants Ra=cd*A*p/(2*m); Rb =g*Ur*cos(theta); N=100; for i=1:N t(i)=i-1; v(i+1)= v0+g*sin(theta)*t(i); s(i+1)= 0.5*g*sin(theta)*t(i).^2; end figure(1) % val1 =abs(find(s==2000)); % v_at2000=v(val1) plot(s,v); grid on title('velocity vs. displacement (Fd=Fr=0)'); ylabel('velocity(m/s)'); xlabel('displacement (m)'); grid on % V(0)=0 for i=1:N t(i)=i-1; V(i+1)= (v0+g*sin(theta)*t(i))-Ra*v(i).^2.*t(i); %0.0000334425= ((cd*A*p)/(2*m))=0.0000334425 S(i+1)= g*sin(theta)*0.5*t(i).^2-Ra*v(i).^2.*0.5*t(i).^2; end figure(2) plot(S,V,'r'); grid on title('velocity vs. displacement (Fr=0, Fd~=0)'); ylabel('velocity(m/s)'); xlabel('displacement (m)'); axis([0 3500 0 70]) for i=1:N t(i)=i-1; V1(i+1)= (v0+g*sin(theta)*t(i))-(cd*A*p/(2*m)*v(i).^2.*t(i))-g*Ur*cos(theta)*t(i); S1(i+1)= g*sin(theta)*0.5*t(i).^2-cd*A*p/(2*m)*v(i).^2.*0.5*t(i).^2-0.5*g*Ur*cos(theta)*t(i).^2; end figure(3) plot(S1,V1,'g'); grid on title('Velocity vs. Displacement(Fr~=0, Fd~=0)'); ylabel('velocity(m/s)'); xlabel('displacement (m)'); axis([0 3500 0 70]) end Results The gravity racer was tested in different conditions where each of the following aspects was present or eliminated. (i) No air resistance, no rolling resistance (ii) No rolling resistance, air resistance included (iii) Air resistance and rolling resistance included Part (i) shows a longer curve spanning a greater displacement with higher speeds. This condition has the gravitational force alone acting on the racer. Ideally, no resistance either or air or on the road surface is experienced. Part (ii) shows a slightly lesser curve than (i). The air resistance is absent, but the racer experiences frictional forces resisting motion on the road surface. It slows down the gravity racer. Part (iii) show a much lesser curve than (ii). Two resistive forces are involved that act against the gravitational forces. These two forces acting against the racer tend to slow down the go-kart even further. The table below shows the displacement and velocity speeds for the three idea situations holding all the standard variables constant. Test No. Displacement (m) Velocity (m/s) (i) 3300 68 (ii) 2700 52 (iii) 2450 49 Discussion Gravity Racing carries a surmountable risk by its little design, engineering, and good practice. This form of risk can be minimized. The racers used in most social amenities for fun and other activities are made using the basis of the technical soapbox regulations. The racers are expected to adhere to this and therefore event in any race worldwide. It is an extreme sport for adults with speeds of a world record more than 85mph. This speed was first recorded in Mont Ventoux during a test drive. Other participants and designers are allowed to engage in designing the racers but for speeds of up to 30mph. It is built to such high technological specifications to make it quite safe for youngsters and ideal to teach vehicle and driving safety. The soapbox is a motorless vehicle that races downhill road either on a timed session or when two or more people are competing. They are built for the purpose of recreation, some gravity racing teams take the sport more seriously and form tournaments. The means of motion is primarily gravity and can achieve speeds of up and above 112km/h or 70miles per hour. Different choices can be made when designing the gravity racer. The following set of choices can be done, and the analysis of the outcome of each is obtained by feeding different constants to a function called racerTester.m where information is obtained about the speeds and distances covered given the different conditions Option 1: steel chassis and spoke wheels Mass 215kg Cd 0.32 A 0.5m2 µr 0.003 The output of the given design when air resistance and rolling resistance are included in the tests results in Option 2: steel chassis and disc wheels Mass 218kg Cd 0.29 A 0.5 µr 0.006 Option 3: carbon fiber monocoque and disc wheels Mass 162kg Cd 0.20 A 0.5 µr 0.006 Option 4: carbon fiber monocoque and spoke wheels Mass 159kg Cd 0.23 A 0.5 µr 0.003 From the options given above, the best design would be obtained using carbon fiber monocoque and spoke wheels. Following different tests carried out varying the different parameter, a design with excellent speeds requires that: - (i) The frontal area has a tiny area tending towards magnitude of zero. (ii) The drag coefficient should be as small as possible. Increasing the air resistance, one can conclude that the gravity racer only moves a short distance at very slow speeds. (iii) Similarly so, the rolling resistance should be a subtle number that resembles the frictional resistance to the road surface should tend towards zero. An increase in rolling resistance lowers the speed and reduces the distance covered by the gravity racer. The slope angle refers to the shape of the terrain where the gravity racer is tested. According to the existing study materials, several tests have been carried out in Mont Ventoux, the USA, and in the UK terrains to test the racer speeds so as to yield a new world speed score. The designers are interested in finding out the highest rate that can be achieved depending on the slopes and the terrain and environment exposed to the go-kart. It is important to know that not all racetracks consist of a constant slope angle. The theta was adjusted to bring out different results from the terrain tested using standard variables. The following slopes are tested Distance Slope angle 0-500m 50 500-1250 20 1250-2000 70 2000-3000 30 The scenario portrayed above could be a rack lane on a different length of the track the slope angles differ hence the racer tends to behave differently regarding the velocity and displacement. Considering this to be a single tournament track with all the forces, that is, gravitational and resistive forces (Fd and Fr) present, the vehicle moves as demonstrated with the data below Distance Slope angle Movement 0-500m 50 Moderate speed 500-1250 20 Slows down to very low speed covering only a short distance 1250-2000 70 Speeds up to very high speed with an ideal situation of up to 4400m displaced. 2000-3000 30 The racer slows down but not as much as with the slope angle 20 with only a short distance covered. The following are screenshots to demonstrate the same The images above show the trend in which the tests carried out on different slope angle behaved. The highest achieved score worldwide is set at 85mph which is relatively high. With more modifications on the chassis material and wheel types, greater speeds can be obtained at terrains with very high slopes. At 70 slope angle, the gravity racer almost rolls back or decelerates. The Matlab scripts above show the commands set to perform the Euler's method of integration as well as the tests to determine the best design and concept to achieve higher speeds. It is evident from all the finding that the slope angle is directly proportional to the speed level, the drag coefficient and rolling resistance coefficient is inversely proportional to the speed as they constitute the resistive forces. The gravitational force is constant at constant mass. Adjusting the mass could either lower or increase the force. As a result, it is prudent to state that the gravitational force is directly proportional to the mass of the body or racer. Conclusion In a nutshell, it is easy to use the gravity racer as a study tool for Newton’s second law of motion. It is the ideal body mass affected by all forces and still manages to propel to speeds even motor vehicle can achieve. One can explore the physics of gravity racing to comprehend the design and course selection influences on the vehicle speed. Using the numerical Euler's method of integration to obtain velocity and displacement from acceleration, it was possible to observe the behavior of the gravity racer by adjusting the standard variables slightly. The case study, however, does not consider other external environmental factors that could hinder movement or beef up the resistive forces. Wind power, rough roads, different temperature values, and weather conditions can cause the gravity racer to slow down or move faster especially with wind power moving in the direction of the gravity racer. Studies and tests are still being carried out to continue searching for the best speeds. References [1] All-American Soapbox Derby 2016 www.soapboxderby.org/aasbd-race-program/ history.aspx [2] Lammey W C 1936 Building your racer for the soap box derby Pop. Mech. 65 754–8 [3] Benacka J 2015 Projectile general motion in a vacuum and a spreadsheet simulation Phys. Educ. 50 58–63 [4] Wade A 2011 Going ballistic: bullet trajectories Undergrad. J. Math. Model. : One + Two 4 1–9 [5] 2014 Speed with Guy Martin: Gravity Racer Channel 4 TV Broadcast 16 November 2014 [6] MapMyFitness, Inc 2016 Map my ride (www.mapmyride.com/) [7] Bodrodz 2012 https://youtu.be/Q1ndyA0UW2w [8] Kyle C R and Weaver M D 2004 Aerodynamics of human-powered vehicles P I Mech. Eng. A-J Pow. 218 141–54 Appendices The free body diagram of a gravity racer is as shown below Read More

Gravity Racing carries a surmountable risk by its little design, engineering, and good practice. This form of risk can be minimized. The racers used in most social amenities for fun and other activities are made using the basis of the technical soapbox regulations. The racers are expected to adhere to this and therefore event in any race worldwide. It is an extreme sport for adults with speeds of a world record of more than 85mph. This speed was first recorded in Mont Ventoux during a test drive.

Other participants and designers are allowed to engage in designing the racers but for speeds of up to 30mph. It is built to such high technological specifications to make it quite safe for youngsters and ideal to teach vehicle and driving safety.The soapbox is a motorless vehicle that races downhill road either on a timed session or when two or more people are competing. They are built for recreation, some gravity racing teams take the sport more seriously and form tournaments. The means of motion is primarily gravity and can achieve speeds of up and above 112km/h or 70miles per hour.

 Different choices can be made when designing the gravity racer. The following set of choices can be done, and the analysis of the outcome of each is obtained by feeding different constants to a function called tracer tester.m where information is obtained about the speeds and distances covered given the different conditionsThe slope angle refers to the shape of the terrain where the gravity racer is tested. According to the existing study materials, several tests have been carried out in Mont Ventoux, the USA, and in the UK terrains to test the racer speeds to yield a new world speed score.

The designers are interested in finding out the highest rate that can be achieved depending on the slopes and the terrain and environment exposed to the go-kart. It is important to know that not all race tracks consist of a constant slope angle. The theta was adjusted to bring out different results from the terrain tested using standard variables.The scenario portrayed above could be a rack lane on a different length of the track the slope angles differ hence the racer tends to behave differently regarding the velocity and displacement.

Considering this to be a single tournament track with all the forces, that is, gravitational and resistive forces (Fd and Fr) present, the vehicle moves as demonstrated with the data below.more modifications on the chassis material and wheel types, greater speeds can be obtained at terrains with very high slopes. At 70 slope angles, the gravity racer almost rolls back or decelerates.The Matlab scripts above show the commands set to perform Euler's method of integration as well as the tests to determine the best design and concept to achieve higher speeds.

It is evident from all the finding that the slope angle is directly proportional to the speed level, the drag coefficient and rolling resistance coefficient is inversely proportional to the speed as they constitute the resistive forces. The gravitational force is constant at constant mass. Adjusting the mass could either lower or increase the force. As a result, it is prudent to state that the gravitational force is directly proportional to the mass of the body or racer.In a nutshell, it is easy to use the gravity racer as a study tool for Newton’s second law of motion.

It is the ideal body mass affected by all forces and still manages to propel to speeds even motor vehicles can achieve. One can explore the physics of gravity racing to comprehend the design and course selection influences on the vehicle speed. Using the numerical Euler's method of integration to obtain velocity and displacement from acceleration, it was possible to observe the behavior of the gravity racer by adjusting the standard variables slightly. The case study, however, does not consider other external environmental factors that could hinder movement or beef up the resistive forces.

Wind power, rough roads, different temperature values, and weather conditions can cause the gravity racer to slow down or move faster especially with wind power moving in the direction of the gravity racer.

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