{
  "id": 357314,
  "title": "Kinematics 3D Projectile Equations",
  "url": "/competitions/tabular-playground-series-oct-2022/discussion/357314",
  "author_name": "moth",
  "post_date": "2022-10-03T23:24:20.380000",
  "votes": 3,
  "comment_count": 0,
  "views": 0,
  "content": "<h1>Projectile motion equations</h1>\n<p>Projectile motion equations can be extremely useful for this competition. One can compute the predicted trajectory of the ball for the future <em>t</em> time steps (assuming the ball is not kicked/disturbed from its trajectory). If the ball's trajectory is aligned towards the goal it will be more likely the player will score.</p>\n<p>The following equations are derived from a projectile in a 3-dimensional space.</p>\n<p><img src=\"https://www.googleapis.com/download/storage/v1/b/kaggle-forum-message-attachments/o/inbox%2F3197853%2F97dcbefb084aa64b8ab9d0e0c063f885%2Fequations.jpg?generation=1664838996195647&amp;alt=media\" alt=\"\"></p>\n<h3>1. Vertical:</h3>\n<p>$$ z(t) = z_{0} + v_{0}.\\sin(\\theta).t + \\frac{1}{2}.g.t^{2} $$</p>\n<p>Where:</p>\n<ul>\n<li><strong>z_{0}:</strong> initial z-coordinate (altitude)</li>\n<li><strong>v_{0}:</strong> initial velocity (module)</li>\n<li><strong>theta:</strong> polar angle</li>\n<li><strong>g:</strong> gravity acceleration</li>\n<li><strong>t:</strong> time</li>\n</ul>\n<h3>2. Horizontal</h3>\n<p>$$ x(t) = x_{0} + (v_{0}.\\cos(\\theta).t).\\cos(\\phi) $$</p>\n<p>$$ y(t) = y_{0} + (v_{0}.\\cos(\\theta).t).\\sin(\\phi) $$</p>\n<p>Where:</p>\n<ul>\n<li><strong>x_{0}:</strong> initial x-coordinate</li>\n<li><strong>y_{0}:</strong> initial y-coordinate</li>\n<li><strong>v_{0}:</strong> initial velocity (module)</li>\n<li><strong>theta:</strong> polar angle</li>\n<li><strong>phi:</strong> azimuthal angle</li>\n<li><strong>t:</strong> time</li>\n</ul>",
  "messages": [
    {
      "id": 1970190,
      "postDate": "2022-10-03T23:24:20.380Z",
      "content": "<h1>Projectile motion equations</h1>\n<p>Projectile motion equations can be extremely useful for this competition. One can compute the predicted trajectory of the ball for the future <em>t</em> time steps (assuming the ball is not kicked/disturbed from its trajectory). If the ball's trajectory is aligned towards the goal it will be more likely the player will score.</p>\n<p>The following equations are derived from a projectile in a 3-dimensional space.</p>\n<p><img src=\"https://www.googleapis.com/download/storage/v1/b/kaggle-forum-message-attachments/o/inbox%2F3197853%2F97dcbefb084aa64b8ab9d0e0c063f885%2Fequations.jpg?generation=1664838996195647&amp;alt=media\" alt=\"\"></p>\n<h3>1. Vertical:</h3>\n<p>$$ z(t) = z_{0} + v_{0}.\\sin(\\theta).t + \\frac{1}{2}.g.t^{2} $$</p>\n<p>Where:</p>\n<ul>\n<li><strong>z_{0}:</strong> initial z-coordinate (altitude)</li>\n<li><strong>v_{0}:</strong> initial velocity (module)</li>\n<li><strong>theta:</strong> polar angle</li>\n<li><strong>g:</strong> gravity acceleration</li>\n<li><strong>t:</strong> time</li>\n</ul>\n<h3>2. Horizontal</h3>\n<p>$$ x(t) = x_{0} + (v_{0}.\\cos(\\theta).t).\\cos(\\phi) $$</p>\n<p>$$ y(t) = y_{0} + (v_{0}.\\cos(\\theta).t).\\sin(\\phi) $$</p>\n<p>Where:</p>\n<ul>\n<li><strong>x_{0}:</strong> initial x-coordinate</li>\n<li><strong>y_{0}:</strong> initial y-coordinate</li>\n<li><strong>v_{0}:</strong> initial velocity (module)</li>\n<li><strong>theta:</strong> polar angle</li>\n<li><strong>phi:</strong> azimuthal angle</li>\n<li><strong>t:</strong> time</li>\n</ul>",
      "rawMarkdown": "# Projectile motion equations\n\nProjectile motion equations can be extremely useful for this competition. One can compute the predicted trajectory of the ball for the future *t* time steps (assuming the ball is not kicked/disturbed from its trajectory). If the ball's trajectory is aligned towards the goal it will be more likely the player will score.\n\nThe following equations are derived from a projectile in a 3-dimensional space.\n\n![](https://www.googleapis.com/download/storage/v1/b/kaggle-forum-message-attachments/o/inbox%2F3197853%2F97dcbefb084aa64b8ab9d0e0c063f885%2Fequations.jpg?generation=1664838996195647&alt=media)\n\n### 1. Vertical:\n\n$$ z(t) = z_{0} + v_{0}.\\sin(\\theta).t + \\frac{1}{2}.g.t^{2} $$\n\nWhere:\n\n- **z_{0}:** initial z-coordinate (altitude)\n- **v_{0}:** initial velocity (module)\n- **theta:** polar angle\n- **g:** gravity acceleration\n- **t:** time\n\n### 2. Horizontal\n\n$$ x(t) = x_{0} + (v_{0}.\\cos(\\theta).t).\\cos(\\phi) $$\n\n$$ y(t) = y_{0} + (v_{0}.\\cos(\\theta).t).\\sin(\\phi) $$\n\nWhere:\n\n- **x_{0}:** initial x-coordinate\n- **y_{0}:** initial y-coordinate\n- **v_{0}:** initial velocity (module)\n- **theta:** polar angle\n- **phi:** azimuthal angle\n- **t:** time",
      "votes": 3
    }
  ],
  "comments": [],
  "raw_markdown_by_id": {
    "1970190": "# Projectile motion equations\n\nProjectile motion equations can be extremely useful for this competition. One can compute the predicted trajectory of the ball for the future *t* time steps (assuming the ball is not kicked/disturbed from its trajectory). If the ball's trajectory is aligned towards the goal it will be more likely the player will score.\n\nThe following equations are derived from a projectile in a 3-dimensional space.\n\n![](https://www.googleapis.com/download/storage/v1/b/kaggle-forum-message-attachments/o/inbox%2F3197853%2F97dcbefb084aa64b8ab9d0e0c063f885%2Fequations.jpg?generation=1664838996195647&alt=media)\n\n### 1. Vertical:\n\n$$ z(t) = z_{0} + v_{0}.\\sin(\\theta).t + \\frac{1}{2}.g.t^{2} $$\n\nWhere:\n\n- **z_{0}:** initial z-coordinate (altitude)\n- **v_{0}:** initial velocity (module)\n- **theta:** polar angle\n- **g:** gravity acceleration\n- **t:** time\n\n### 2. Horizontal\n\n$$ x(t) = x_{0} + (v_{0}.\\cos(\\theta).t).\\cos(\\phi) $$\n\n$$ y(t) = y_{0} + (v_{0}.\\cos(\\theta).t).\\sin(\\phi) $$\n\nWhere:\n\n- **x_{0}:** initial x-coordinate\n- **y_{0}:** initial y-coordinate\n- **v_{0}:** initial velocity (module)\n- **theta:** polar angle\n- **phi:** azimuthal angle\n- **t:** time"
  }
}