d4t_formulas/assets/formulas/formulas.d4rt

[
  // Free fall distance (vertical)
  {
    "name": "Free Fall Distance",
    "description": r"""
Calculates vertical displacement under constant gravity

$$h = \frac{1}{2}gt^2$$

Where:
- $g$: Gravitational acceleration $9.81\ \mathrm{m/s^2}$ on Earth
- $t$: Time in free fall (seconds)

![Free Fall Diagram](https://altcalculator.com/wp-content/uploads/2023/08/Free-Fall.png)""",
    "input": [
      {"name": "t", "unit": "second"},
      {"name": "g", "unit": "meters per second"}
    ],
    "output": {"name": "h", "unit": "meter"},
    "d4rtCode": "h = 0.5 * g * pow(t, 2);",
    "tags": ["physics", "kinematics"]
  },

  // Newton's Law of Universal Gravitation
  {
    "name": "Gravitational Force",
    "description": r"""
Newton's law of universal gravitation

$$F = G\frac{m_1m_2}{r^2}$$

Where:
- $G$: Gravitational constant $6.674\times 10^{-11}\ \mathrm{N\cdot m^2/kg^2}$
- $m_1, m_2$: Masses of two objects
- $r$: Distance between centers of masses

![Gravitation](https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/NewtonsLawOfUniversalGravitation.svg/1200px-NewtonsLawOfUniversalGravitation.svg.png)""",
    "input": [
      {"name": "m1", "unit": "kilogram"},
      {"name": "m2", "unit": "kilogram"},
      {"name": "r", "unit": "meter"}
    ],
    "output": {"name": "F", "unit": "newton"},
    "d4rtCode": "F = (6.67430e-11 * m1 * m2) / pow(r, 2);",
    "tags": ["physics", "astronomy", "gravity"]
  },

  // Projectile Motion Range
  {
    "name": "Projectile Range",
    "description": r"""
Calculates horizontal distance of projectile motion

$$R = \frac{v^2 \sin(2\theta)}{g}$$

Where:
- $v$: Initial velocity
- $\theta$: Launch angle
- $g$: Gravitational acceleration

![Projectile Motion](https://upload.wikimedia.org/wikipedia/commons/thumb/5/52/Projectile_motion_diagram.png/800px-Projectile_motion_diagram.png)""",
    "input": [
      {"name": "v", "unit": "meters per second"},
      {"name": "a", "unit": "degree"}
    ],
    "output": {"name": "R", "unit": "meter"},
    "d4rtCode": """
       var radians = a * (pi / 180);
       R = (pow(v, 2) * sin(2 * radians)) / 9.80665;
    """,
    "tags": ["physics", "kinematics", "projectile"]
  },

  // Newton's Second Law
  {
    "name": "Newton's Second Law",
    "description": r"""
Force equals mass times acceleration

$$F = m \cdot a$$

Where:
- $m$: Mass of object ($\mathrm{kg}$)
- $a$: Acceleration ($\mathrm{m/s^2}$)

![Newton's Second Law](https://upload.wikimedia.org/wikipedia/commons/thumb/7/73/Newtonslawsofmotion.jpg/800px-Newtonslawsofmotion.jpg)""",
    "input": [
      {"name": "m", "unit": "kilogram"},
      {"name": "a", "unit": "meters per square second"}
    ],
    "output": {"name": "F", "unit": "newton"},
    "d4rtCode": "F = m * a;",
    "tags": ["physics", "mechanics", "newton"]
  },

  // Centripetal Force
  {
    "name": "Centripetal Force",
    "description": r"""
Force required to keep an object moving in circular motion

$$F = \frac{mv^2}{r}$$

Where:
- $F$: Centripetal force (Newtons)
- $m$: Mass of object (kilograms)
- $v$: Velocity (m/s)
- $r$: Radius of circular path (meters)

This force acts toward the center of the circle.""",
    "input": [
      {"name": "m", "unit": "kilogram"},
      {"name": "v", "unit": "meters per second"},
      {"name": "r", "unit": "meter"}
    ],
    "output": {"name": "F", "unit": "newton"},
    "d4rtCode": "F = (m * pow(v, 2)) / r;",
    "tags": ["physics", "circular motion", "centripetal"]
  },

  // Wave Equation
  {
    "name": "Wave Equation",
    "description": r"""
Relationship between wave speed, frequency, and wavelength

$$v = f\lambda$$

Where:
- $v$: Wave speed (m/s)
- $f$: Frequency (Hertz)
- $\lambda$: Wavelength (meters)

This applies to all types of waves including sound and light.""",
    "input": [
      {"name": "f", "unit": "hertz"},
      {"name": "lambda", "unit": "meter"}
    ],
    "output": {"name": "v", "unit": "meters per second"},
    "d4rtCode": "v = f * lambda;",
    "tags": ["physics", "waves", "frequency"]
  },

  // Pythagorean Theorem
  {
    "name": "Pythagorean Theorem",
    "description": r"""
Fundamental relation in Euclidean geometry among the three sides of a right triangle

$$a^2 + b^2 = c^2$$

Where:
- $a$, $b$: Legs of the right triangle
- $c$: Hypotenuse of the right triangle

The square of the hypotenuse is equal to the sum of squares of the other two sides.""",
    "input": [
      {"name": "a", "unit": "meter"},
      {"name": "b", "unit": "meter"}
    ],
    "output": {"name": "c", "unit": "meter"},
    "d4rtCode": "c = sqrt(pow(a, 2) + pow(b, 2));",
    "tags": ["trigonometry", "geometry", "pythagorean"]
  },

  // Sine Rule
  {
    "name": "Sine Rule",
    "description": r"""
Relationship between the sides and angles of any triangle

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Where:
- $a$, $b$, $c$: Sides of the triangle
- $A$, $B$, $C$: Angles opposite to sides $a$, $b$, $c$ respectively

This rule is useful for solving triangles when certain combinations of angles and sides are known.""",
    "input": [
      {"name": "a", "unit": "meter"},
      {"name": "A", "unit": "degree"},
      {"name": "B", "unit": "degree"}
    ],
    "output": {"name": "b", "unit": "meter"},
    "d4rtCode": """
       var angleARad = A * (pi / 180);
       var angleBRad = B * (pi / 180);
       b = (a * sin(angleBRad)) / sin(angleARad);
    """,
    "tags": ["trigonometry", "triangle", "sine"]
  },

  // Cosine Rule
  {
    "name": "Cosine Rule",
    "description": r"""
Generalization of the Pythagorean theorem for any triangle

$$c^2 = a^2 + b^2 - 2ab\cos(C)$$

Where:
- $a$, $b$, $c$: Sides of the triangle
- $C$: Angle opposite to side $c$

This rule relates all three sides of a triangle to one of its angles.""",
    "input": [
      {"name": "a", "unit": "meter"},
      {"name": "b", "unit": "meter"},
      {"name": "C", "unit": "degree"}
    ],
    "output": {"name": "c", "unit": "meter"},
    "d4rtCode": """
       var angleCRad = C * (pi / 180);
       c = sqrt(pow(a, 2) + pow(b, 2) - 2*a*b*cos(angleCRad));
    """,
    "tags": ["trigonometry", "triangle", "cosine"]
  },

  // Trigonometric Identity
  {
    "name": "Trigonometric Identity",
    "description": r"""
Fundamental Pythagorean identity in trigonometry

$$\sin^2(\theta) + \cos^2(\theta) = 1$$

Where:
- $\theta$: Any angle in radians or degrees

This identity is derived from the Pythagorean theorem applied to the unit circle.""",
    "input": [
      {"name": "theta", "unit": "degree"}
    ],
    "output": {"name": "result", "unit": "scalar"},
    "d4rtCode": """
       var thetaRad = theta * (pi / 180);
       result = pow(sin(thetaRad), 2) + pow(cos(thetaRad), 2);
    """,
    "tags": ["trigonometry", "identity", "sine", "cosine"]
  },

  // Gravitational Potential Energy
  {
    "name": "Gravitational Potential Energy",
    "description": r"""
Energy possessed by an object due to its position in a gravitational field

$$PE = mgh$$

Where:
- $m$: Mass of object (kilograms)
- $g$: Gravitational acceleration ($9.81\ \mathrm{m/s^2}$ on Earth)
- $h$: Height above reference point (meters)

This energy increases with height and can be converted to kinetic energy.""",
    "input": [
      {"name": "m", "unit": "kilogram"},
      {"name": "h", "unit": "meter"},
      {"name": "g", "unit": "meters per square second"}
    ],
    "output": {"name": "PE", "unit": "joule"},
    "d4rtCode": "PE = m * g * h;",
    "tags": ["physics", "energy", "mechanics", "gravity"]
  },

  // Linear Momentum
  {
    "name": "Linear Momentum",
    "description": r"""
Quantity of motion possessed by a moving object

$$p = mv$$

Where:
- $p$: Momentum ($\mathrm{kg \cdot m/s}$)
- $m$: Mass (kilograms)
- $v$: Velocity (m/s)

Momentum is conserved in isolated systems.""",
    "input": [
      {"name": "m", "unit": "kilogram"},
      {"name": "v", "unit": "meters per second"}
    ],
    "output": {"name": "p", "unit": "kilogram meter per second"},
    "d4rtCode": "p = m * v;",
    "tags": ["physics", "mechanics", "momentum"]
  },

  // Volume of Sphere
  {
    "name": "Volume of Sphere",
    "description": r"""
Volume enclosed by a sphere

$$V = \frac{4}{3}\pi r^3$$

Where:
- $V$: Volume (cubic meters)
- $r$: Radius (meters)
- $\pi$: Pi ($\approx 3.14159$)

The volume is proportional to the cube of the radius.""",
    "input": [
      {"name": "r", "unit": "meter"}
    ],
    "output": {"name": "V", "unit": "cubic meter"},
    "d4rtCode": "V = (4.0/3.0) * pi * pow(r, 3);",
    "tags": ["geometry", "sphere", "volume"]
  },

  // Surface Area of Sphere
  {
    "name": "Surface Area of Sphere",
    "description": r"""
Total surface area of a sphere

$$A = 4\pi r^2$$

Where:
- $A$: Surface area (square meters)
- $r$: Radius (meters)
- $\pi$: Pi ($\approx 3.14159$)

The surface area is four times the area of a circle with the same radius.""",
    "input": [
      {"name": "r", "unit": "meter"}
    ],
    "output": {"name": "A", "unit": "square meter"},
    "d4rtCode": "A = 4 * pi * pow(r, 2);",
    "tags": ["geometry", "sphere", "surface area"]
  },

  // Volume of Cylinder
  {
    "name": "Volume of Cylinder",
    "description": r"""
Volume enclosed by a cylinder

$$V = \pi r^2 h$$

Where:
- $V$: Volume (cubic meters)
- $r$: Radius of base (meters)
- $h$: Height (meters)
- $\pi$: Pi ($\approx 3.14159$)

The volume is the area of the base times the height.""",
    "input": [
      {"name": "r", "unit": "meter"},
      {"name": "h", "unit": "meter"}
    ],
    "output": {"name": "V", "unit": "cubic meter"},
    "d4rtCode": "V = pi * pow(r, 2) * h;",
    "tags": ["geometry", "cylinder", "volume"]
  },

  // Surface Area of Cylinder
  {
    "name": "Surface Area of Cylinder",
    "description": r"""
Total surface area of a cylinder (including top and bottom)

$$A = 2\pi r(r + h)$$

Where:
- $A$: Total surface area (square meters)
- $r$: Radius of base (meters)
- $h$: Height (meters)
- $\pi$: Pi ($\approx 3.14159$)

This includes the lateral surface plus the two circular ends.""",
    "input": [
      {"name": "r", "unit": "meter"},
      {"name": "h", "unit": "meter"}
    ],
    "output": {"name": "A", "unit": "square meter"},
    "d4rtCode": "A = 2 * pi * r * (r + h);",
    "tags": ["geometry", "cylinder", "surface area"]
  },

  // Volume of Cone
  {
    "name": "Volume of Cone",
    "description": r"""
Volume enclosed by a cone

$$V = \frac{1}{3}\pi r^2 h$$

Where:
- $V$: Volume (cubic meters)
- $r$: Radius of base (meters)
- $h$: Height (meters)
- $\pi$: Pi ($\approx 3.14159$)

The volume is one-third the volume of a cylinder with the same base and height.""",
    "input": [
      {"name": "r", "unit": "meter"},
      {"name": "h", "unit": "meter"}
    ],
    "output": {"name": "V", "unit": "cubic meter"},
    "d4rtCode": "V = (1.0/3.0) * pi * pow(r, 2) * h;",
    "tags": ["geometry", "cone", "volume"]
  },

  // Volume of Cube
  {
    "name": "Volume of Cube",
    "description": r"""
Volume enclosed by a cube

$$V = s^3$$

Where:
- $V$: Volume (cubic meters)
- $s$: Side length (meters)

The volume is the cube of the side length.""",
    "input": [
      {"name": "s", "unit": "meter"}
    ],
    "output": {"name": "V", "unit": "cubic meter"},
    "d4rtCode": "V = pow(s, 3);",
    "tags": ["geometry", "cube", "volume"]
  },

  // Surface Area of Cube
  {
    "name": "Surface Area of Cube",
    "description": r"""
Total surface area of a cube

$$A = 6s^2$$

Where:
- $A$: Total surface area (square meters)
- $s$: Side length (meters)

The surface area is six times the area of one face.""",
    "input": [
      {"name": "s", "unit": "meter"}
    ],
    "output": {"name": "A", "unit": "square meter"},
    "d4rtCode": "A = 6 * pow(s, 2);",
    "tags": ["geometry", "cube", "surface area"]
  },

  // Perimeter of Rectangle
  {
    "name": "Perimeter of Rectangle",
    "description": r"""
Total distance around a rectangle

$$P = 2(l + w)$$

Where:
- $P$: Perimeter (meters)
- $l$: Length (meters)
- $w$: Width (meters)

The perimeter is twice the sum of length and width.""",
    "input": [
      {"name": "l", "unit": "meter"},
      {"name": "w", "unit": "meter"}
    ],
    "output": {"name": "P", "unit": "meter"},
    "d4rtCode": "P = 2 * (l + w);",
    "tags": ["geometry", "rectangle", "perimeter"]
  },

  // Perimeter of Triangle
  {
    "name": "Perimeter of Triangle",
    "description": r"""
Total distance around a triangle

$$P = a + b + c$$

Where:
- $P$: Perimeter (meters)
- $a, b, c$: Side lengths (meters)

The perimeter is the sum of all three sides.""",
    "input": [
      {"name": "a", "unit": "meter"},
      {"name": "b", "unit": "meter"},
      {"name": "c", "unit": "meter"}
    ],
    "output": {"name": "P", "unit": "meter"},
    "d4rtCode": "P = a + b + c;",
    "tags": ["geometry", "triangle", "perimeter"]
  }
]