1. Basic Physics for the VEX Game
Trajectory
Basic Physics for the VEX Game
Draft Updated: 9/27/2015

2. Trajectory
A trajectory or flight path is the path that a moving object follows through space as a function of time.
The object might be a projectile or a satellite.
A object in motion through space (for example a thrown baseball) may be called a projectile
A trajectory can be described using equations of motion

3. Trajectory
Equations of motion in mathematical physics, are equations that describe the behaviour of a physical system in terms of its motion as a function of time
Newton's theory (laws) developed into the branch of theoretical physics known as classical mechanics. It employs the mathematics of differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change.

4. Trajectory

5. Trajectory

6. Trajectory sin 2 R = ————— g sin 2g g - Vi Vi h = ——————- Vi h R Vi -
The range, R, is the greatest distance the object travels along the x-axis in the I sector. 2 2 Vi
sin ⩉i
h = ——————- Vi 2g h
The height, h, is the greatest parabolic height said object reaches within its trajectory ⩉i
g -
The g is the respective gravitational pull on the object R
⩉i -
The initial angle, θi, is the angle at which said object is released.
Vi -
The initial velocity, vi, is the speed at which said object is launched from the point of origin.

7. Basic Trigonometry
In mathematics, the trigonometric functions (also called the circular functions) are functions of an angle.
They relate the angles of a triangle to the lengths of its sides.
Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.

8. Basic Trigonometry B h
(hypotenuse) a
(opposite) A b C
(adjacent)

9. Basic Trigonometry
In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a relation in Euclidean geometry among the three sides of a right triangle.
It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation"

10. Basic Trigonometry a2 + b2 = c2 Pythagorean theorem:
Where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides.

11. Angular to Linear Velocity
The Angular to Linear Velocity formula is :
v = r × ω
Where:
v: Linear velocity, in m/s
r: Radius, in meter
ω: Angular velocity, in rad/s

12. Angular to Linear Velocity
The radian (rad) is the standard unit of angular measure, used in many areas of mathematics.
An angle's measurement in radians is numerically equal to the length of a corresponding arc of a unit circle;
One radian is just under 57.3 degrees (when the arc length is equal to the radius)

13. Angular to Linear Velocity

14. Angular to Linear Velocity
The RPM to Linear Velocity formula is :
  v = r × RPM ×
Where:
v: Linear velocity, in m/s
r: Radius, in meter
RPM: Angular velocity, in RPM (Rounds per Minute)
Handy Calculator:

15. Launch Velocity
The launch velocity of a projectile can be calculated from the range if the angle of launch is known. It can also be calculated if the maximum height and range are known, because the angle can be determined.

16. Launch Velocity
For Calculator see:

17. Launch Velocity
Will it clear the goal?

18. Trajectory Analysis
Using Logger Pro

19. Trajectory Example

20. Trajectory Example

21. Trajectory Example
Data points we really want for our ball shooter, are:
take off angle
take off velocity
range travelled
height reached
We want to calculate these at V0x and V0y this is for t = 0
Take off angle: V0
V0y ( )
V0y A0
= tanh
= tanh(1.842/3.141) = deg A0
V0x
V0x

22. Trajectory Example The take off velocity can be calculated as:
V0 = SQRT(V0x2 + V0y2)
V0 = SQRT( )
V0 = 3.641m/s

23. Using LoggerPro
Video Analysis

24. Video Analysis Production and analysis of your own movie
You will need either a digital video camera or a digital still camera set to “movie mode”. Keep the following tips in mind when you shoot your movie.
Using a smart phone will work or the camera in your laptop
It is best to have a plain background that provides sufficient contrast with the projectile. Good lighting is essential
Set up the camera on a tripod so that it is looking square at the background, and so that the plane of motion is perpendicular to the view
Position the camera as far from the plane of motion as is practical to reduce problems with scaling and parallax. Use the zoom feature to fill the screen with the motion.

25. Video Analysis Production and analysis of your own movie
The object used for scaling must be in the same plane as the motion of the projectile (see Figure 2).
Video analysis can also be performed on iOS devices using Video Physics for iOS.

26. Video Analysis Production and analysis of your own movie
Once you have shot your movie, use the directions that accompany your camera to transfer the video clip to the computer you will use for the analysis. If you have captured more video than you need and your movie is much too large, you can use video editing software (e.g. QuickTime Pro or iMovie) to edit the clip down to a more manageable length.

27. Video Analysis Analyzing there Movie Start Logger Pro
Go to Insert -> movie and import the movie of your trajectory
Make the movie window large enough to easily see the projectile
Enable Video Analysis by clicking on the button in the lower-right corner. This brings up a toolbar with a number of buttons (see Figure 1).

28. Video Analysis

29. Video Analysis Analyzing there Movie
Click the Set Origin button (third from top), then click in the movie frame to set the location of the origin. If needed, this coordinate system can be rotated by dragging the yellow dot on the horizontal axis.
Click the Set Scale button (fourth from top), then drag across an object of known length in the movie. In this movie, the object of known length is the 2 m stick on the floor. When you release the mouse button, enter the length of the object; be sure the units are correct.
Use the forward and back movie buttons to advance the movie until the ball is released from the shooter’s hands. Next to the button you used to enable analysis is the Sync Movie to Graph button. Click this button, then enter 0 in the graph time window. Select Use This Synchronization in Video Capture.

30. Video Analysis Analyzing there Movie
Now click the Add Point button (second from the top). Decide where on the object you will mark its location (center, top, other) and then click the object in the movie. Important: Be consistent in your marking.
Each time you mark the object’s location, the movie advances one frame. Depending on the frame rate, you may choose to mark the position every other frame. Notice that data are being plotted on the graph.
Continue this process as long as is desired. Should you wish to edit a point, click the Select Point button (top). This allows you to move or delete a mis marked point.
Select the graph window. Logger Pro defaults to display both the x and y positions of the object as a function of time. You may find it easier to examine the position-time behavior of just one of these components at a time.

31. Video Analysis Evaluating the Data
Examine the graph of x-position vs. time. If it appears to be linear, fit a straight line to your data. If the slope of the graph appears to change abruptly, fit separate straight lines to each portion of the graph that appears to be linear.
Linear Fit Tool

32. Video Analysis Evaluating the Data
Now, examine the graph of y-position vs. time. Fit an appropriate curve to this graph (or to each portion of the graph). Write the equation that describes the y-position vs. time behavior of the ball in the first segment; be sure to include units.
Curve Fit Tool

33. Linear Velocity

34. Angular to Linear Velocity
The RPM to Linear Velocity formula is :
  v = r × RPM ×
Where:
v: Linear velocity, in m/s
r: Radius, in meter
RPM: Angular velocity, in RPM (Rounds per Minute)
Handy Calculator:

35. Linear Velocity - VEX 2.75” wheel 0.06985m 4.0” wheel 50g 0.1016m 90g
50g

36. Linear Velocity - VEX Output Stage Driving Gear
Output Stage Driven Gear
Output Speed (RPM)
Output Stall Torque (N*m)
IME Ticks per Revolution
Standard Motor 393 Gearing
10t
32t
100
1.67
627.2
High Speed Option
(included with Motor 393)
14t
28t
160
1.04
392
Turbo Gear Set
(sold separately)
18t
24t
240
0.7

37. Linear Velocity - VEX Wheel RPM V (m/s) 4” (0.1016m) 100 1.06396m/s
160
1.7023m/s
2.75” ( m)
0.7315m/s
1.1704m/s

38. Linear Velocity - VEX
Gear Teeth Count 12 36 60 84

39. Linear Velocity - VEX 12 36 60 84 1:1 1:3 1:5 1:7 3:1 1:1.67 1:2.3 5:1
1.67:1
1:1.4
7:1
2.3:1
1.14:1

40. Linear Velocity - VEX
HS Gear Teeth Count 12 36 60

41. Linear Velocity - VEX High Strength Gear Kit 12 36 60 1:1 1:3 1:5 3:1
1:1.67
5:1
1.67:1

42. Maximum ideal gear ratio possible for a wheel
Linear Velocity - VEX
Maximum ideal gear ratio possible for a wheel
F = m x a Force required to ‘lift’ weight
T = F x d Torque required to turn weight
For 4” (0.1016m) wheel weighing 90gr
F = 0.09 x 9.81 = N
T = x = 0.089Nm
Max gear ratio at 100rpm motor — but that is at stall torque!
1.67 / = 1:18 gear ratio

43. Solving the Trajectory For Bankshot Game

44. Bankshot Game tan ⩉ = ——— = 44deg Vi 178mm 559mm 395mm 168mm
( )
tan ⩉ = ———
= 44deg
( )

45. Bankshot Game With the following values:
⩉i = 65 Deg
The take of Velocity Vi will be approximately: 3.3m/s
Yet a VEX IQ motor has a maximum RPM = 120, and 100mm diameter wheel, this translates into a linear velocity = 1.256m/s
In this case we would need to speed up the motor rotation by at least: 3.3m/s / m/s = 3 times

46. Bankshot Game We have the following gears available in VEX IQ: 12 2436 48 60
So we could use gear train to achieve a 1:3 gear ratio and 1:3 speed up, however we need to take in consideration the torque reduction!

47. Solving the Trajectory For Nothing but Net