Project: The Motion of a Golf Ball

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authors: Celso R. C. Rêgo

Usually, people and especially students, have some issues with the concept that the only force acting upon an upward-moving projectile is gravity. Their conception of motion prompts them to think that if an object moves in a given direction, then there must be a force in that direction, e.g., if a particle is moving upward and rightward, there must be both an upward and rightward force. Probably, they believe it due to their intuition based on their daily lives, where the air-resistance, often termed drag, might not be negligible. In this project, we combine Python with Simstack workflow framework to understand how we handle the projectile motions when drag effects are important.

Naturally, it is not complicated to add the drag force in the equations for a projectile motion, but solving them analytically for the position and velocity as functions of time can get a bit tricky. Fortunately, it is relatively easy to make precise and accurate numerical solutions using Python's Scipy library.

This project aims to show how the Projectile motion experienced by a Golf ball with mass $M$ $(kg)$ and radius $r (m)$ projected near the Earth's surface moves along a curved path under the action of gravity, $g=9.81$ $(m/s^2)$ when the effects of air-resistance and lift are assumed or neglected. Beyond the physics problems we aim to solve, we also want to highlight the MultipleOf and ForEach loop control features in the Simstack workflow framework. With only these features, we will submit several setups at once for a given simulation protocol, which might be computed in series using MultipleOf or parallel with ForEach, and thus get the desired physical properties.

This tutorial demonstrates step-by-step through the Motion of a Golf ball project how to run and build simple and complex workflows. With these examples, we will learn how to iterate over parameter ranges using ForEach loop control. Additionally, we explain how workflows may be branched through If loop control. The tutorial has a section with a breve introduction regards to physics of the problem. The remaining sections explore how to use Simstack features to understanding the nature of the problem.

In this workflow, we will be able to:

1. Understanding the connection of acceleration due to gravity, range, maximum height, and trajectory properties of a projectile motion.
2. Determine the velocity of a particle at different points in its trajectory.
3. Apply the principle of independence of motion to solve projectile motion problems when the effects of air-resistance are assumed or neglected.

Theoretical Solution

The proper way to modeling the Projectile Motion is to split it into two independent motions, i.e., horizontal (x) and vertical (y). The magnitudes of the components of the velocity $V$ are $V_x = V cos(\theta)$ and $V_y = V sin(\theta)$ where $V$ is the magnitude of the velocity and $\theta$ is its angle direction, as depicted in Fig 1. A golf ball's trajectory is affected directly by gravity, air resistance, and rotation (lift). We illustrate on the right of Fig 2 the forces acting on the ball. This project will explore four scenarios: a smooth ball, smooth ball + drag, golf ball + drag, and golf ball + drag + lift.

Fig 1 The split of a projectile motion into two independent one-dimensional motions along the vertical and horizontal axes. The horizontal range $d$ ( the blue line) is maximum distance traveled on the $x$ coodinate when it returns to its initial height ($y=0$).

2D General Equations

$F_x = - F^d_x + F^l_x$

$F_y = -F_g - F^d_x + F^l_y$

$F_g$ is the force of gravity that acts at all times on all objects near Earth.

The drag force or air resistance $F^d = \frac{1}{2}\rho_f A C_D|v|^2$ refers to forces acting opposite to the relative motion of the ball moving concerning a surrounding fluid (air in this case). Here, $C_D$ is the drag coefficient, ${\rho}$ is the air density, ${A}$ represents the cross-sectional area of the ball. The drag coefficient is not constant, it decreases as velocity increases.

$F^l = \frac{1}{2}\rho_f A C_L|v|^2$ is the lift force stemming from the rotation of the ball (the Magnus-effect) and is normal to $v$. With the given direction, the ball rotates counter-clockwise (backspin), where $C_L$ is the lift coefficient.

Projectile-motion WaNo

The Workflow building blocks within the Simstack Workflow framework are composed of Workflow Active Nodes (WaNos), which are XML files combined with scripts defining the expected input and output. As pointed out above, we want to understand the physics of a Projectile motion accounting or not drag effects; for that, we built a WaNo as shown in Fig 2, where only the relevant parameters are exposed.

Fig 2 On the left-hand side is depicted the Projectile_motion WaNo. Outlined in blue we expose the most relevant physical parameters of the projectile motion problem. On the right-hand side, we depict some of the possible forces acting on the golf ball.

1. Python Setup

To get this workflow up running on your available computational resources, make sure to have the below libraries installed on Python 3.6 or newer.

1. Numpy, os, sys, csv, yaml
2. scipy
3. matplotlib

2. Inputs parameters with MultipleOf feature

Parameter	variable type
System	Boolean
x0(m)	Float
y0(m)	Float
v0(m/s)	Float
Mass (kg)	Float
Angle(°)	Float
Radius(m)	Float
label	String

The list above displays the Projectile_motion WaNo parameters with the respective variable types and physical units. Here, $x_0$ and $y_0$ are the initial positions of the projectile in the horizontal and vertical axes. $v_0$ is the initial velocity. $Mass$ is the ball's mass with a given Radius, and the label variable is a string to assign the chosen set of the variables. The System flag adds the desired scenario, and the equations of motion are solved numerically using the solve_ip from_ scipy_ library.

The set of the exposed parameters in this WaNo allows us to change the python script's inputs embedded on it. Based on the WaNoDropDown and MultipleOf features, this interface may submit many serial tasks at once, as shown on the evaluated three initial angle values on the right-hand side of Fig 2. The outcomes follow the numerical solutions for the projectile motion within the chosen scenario.

3. Outputs

This WaNo will generate PROJOUT.yml and PROJDATA.yml files. The table below shows the keys contained in each one, and later on, we will use these keys to inquire about their values.

PROJOUT.yml	PROJDATA.yml
xmax maximum range	x position
ymax maximum height	y position
time to target	vx velocity
time to highest point	vy velocity
Step ii label

4. Auxiliary WaNos

The set of Auxiliary WaNos will be extensively used and reused in all upcoming workflows. They will be responsible for managing some worflow data. As shown in Fig 3, Range-It, Plot-Figures and Table-Generator, will be used to pass a variable at the beginning of the workflow, inquire variables of a loaded file, and plot figures.

Range-It creates a Float or integer list, which will pass to the Projectile-motion WaNo inside the ForEach loop control, explained in the next step.

The Table-Generator WaNo generates table files in a csv and yml formats for a given set of variables inquired from a loaded file.

The Plot-Figures WaNo will make a plot of the inquired data. This WaNo allows us to switch between Same-graph (plot several curves in the same figure) and Subplot modes (plot each curve in a different subplot ).

Fig 3 The upper two panels exhibit the Float and Int modes available on the Range-It WaNo. The below two panels display the Same-graph and Subplot modes. Each mode in this WaNo allows us to inquire about the variables from Projectile-motion and plot them.

The outputs of the WaNo Plot-Figures in Fig 3 might be _Plot-Figure.png _ and Plot-subplot.png . Click on Fig 3 to see more details about their inputs.

5. Workflow with Projectile-motion and Plot-Figures WaNos

Fig 4 The workflow above is composed of Projectile-motion, and Plot-Figures WaNos. Step 3 shows the figure as one of the possible outputs of this workflow.

Fig 4 shows the workflow named as Proj-motion, which compares the drag effect acting on a smooth ball.

Running this Workflow

• 1. Drag and drop the Proj-motion WaNo from the top left menu to the SimStack canvas as pointed by the blue arrow on panel Step 1 in Fig 4.
• 2. In this case, we set the Angle parameter to 25(°) for two different System scenarios (smooth ball and smooth ball + drag ), we kept the other parameters as their default values.
• 3. Repeat Step 1 for auxiliary Plot-Figures WaNo connecting it below the Proj-motion. Load the PROJOUT.yml file field in the Input-File field, then click on the option Same-graph, the click will trigger the options to be filled. In this case, you should set the title, labels, and variables (data), which will show up in the output figure.
• 4. Name your workflow with Ctrl+S, and running it with Ctrl+R command.
• 5. The Step 3 of Fig 4 shows that by choosing the Browser Directory with a double click in the green folder (Jobs & Workflows tab) of the workflow, you will be able to click on Plot-Figure.png and see the figure comparing the x and y coordinates of the smooth ball under or not of air resistance effect.

6. A slightly complex workflow using the ForEach feature

In this Workflow, we want to explore the scenario where the system under study has multiples initial velocity (v_0) values, and we want to investigate the dependence of maximum height ymax and time to target variables in terms of maximum range xmax. For this example, the chosen system is golf ball + drag + lift.

Fig 5 shows the workflow, a workflow composed of four WaNos and the ForEach loop control. The blue arrows refer to the input parameters of each WaNo. The red arrow in Step 1 shows how to fill the field responsible for passing the list of values from Range-It WaNo to the ForEach. The red arrow in Step 3 points out the assignment of the ForEach_Iterator to the initial velocity (v_0) variable. The red arrow in Step 5 shows the path to import all the files PROJOUT.yml of each initial velocity value. The last red arrow in Step 6 indicates the tab where we must browser to access the Plot-subplot.png figure.

Running this Workflow

• 1. Drag and drop the Range-It WaNo from the top left menu to the SimStack canvas as pointed by the blue arrow on panel Step 1 in Fig 5. There are 20 different values for initial velocity in this scenario, varying from 20 to 70 (m/s).
• 2. Drag and drop the ForEach loop control from below right and insert the Projectila-motion WaNo inside it. In the sequence, assign the ${ForEach_Iterator} according to the Step 3 of Fig 5.
• 3. Drag and drop the Table-Generator WaNo from the top left menu to the SimStack below to ForEach loop control. Fill up the fields of Table-Generator as shown in Step 4 of Fig 5. In this part, you also should import the files from where the information will be extracted, in this case PROJOUT.yml as depicted in Step 5.
• 4. Drag and drop the Plot-Figures WaNo from the top left menu to the SimStack below to Table-Generator, click on the option Subplot. In this case, you should set the title, labels, and variables (data), which will show up in the output figure. Fill up the fields according to the Step 6 of Fig 5.
• 5. Name your workflow with Ctrl+S, and running it with Ctrl+R command.
• 6. The last step in Fig 5 shows that by choosing the Browser Directory with a double click in the green folder (Jobs & Workflows tab) of the workflow, you will be able to click on Plot-subplot.png and see the subplots comparing the dependence of maximum height ymax and time to target variables in terms of maximum range xmax.

7. Branched Workflows using the If feature

This part will explain how to preventing unphysical results using the If loop control, which essentially branches the workflow. In the Projectile-motion WaNo the options golf ball + drag and golf ball + drag + lift in the System field are only valid for initial velocities v0(m/s) between 13.7 and 88.1 m/s. This constraint occurs due to the dependence of the drag and lift coefficients, which are functions of the initial velocities and spin of the golf ball, as pointed out in the beginning. In this case, we are keeping the spin constant. Then only the velocity will be considered.

Fig 6 shows a branched workflow, which prevents unphysical results for a specific variable. The black arrows in both steps point from the variable val_v0 value to two different scenarios inside the If loop control.

Fig 6 exhibits the outcomes from this example. The workflow left, and the right sides display the two possible scenarios for this case. Runs the workflow composed by the Projectile-motion and Plot-Figures or runs Stop WaNo, which prints out a message on the Stop-msg file.

Running this Workflow

• 1. Drag and drop the Variable control from the bottom left menu to the SimStack canvas and setup it as shows Fig 6.
• 2. Drag and drop the If control bottom left menu and insert on the left-hand side the workflow composed by the Projectila-motion and Plot-Figures WaNos. Next, we make the appropriate setup for them. If this part is true, it must generate the expected output files for each WaNo as explained in section 5.
• 3. Drag and drop the auxiliary Stop WaNo from the bottom left menu inside the right side of the If loop control. If this part is true, it must generate the Stop-msg file.
• 5. Name your workflow with Ctrl+S, and running it with Ctrl+R command.
• 6. A double click in the green folder (Jobs & Workflows tab) of the workflow will allow us to check the outputs according to the chosen if condition.

Final Remarks

Running this project within Simstack saves time, and we avoid adding more code lines to our python script. For instance, to get the figure in Step 6, we would have to add a for loop in the python script to be executed in a serial version, unless you want to make an additional effort to parallelize this task. On the other hand, Simstack promptly runs it in parallel in the available computational resources.