Universe

Universe is a two-body gravitational simulation that models the motion of two masses under mutual gravitational attraction. The user inputs mass, initial velocity, launch angle, and starting coordinates for each body, along with the gravitational constant, giving full control over the simulated physics. Velocities are decomposed into x and y components using trigonometry, and each time step applies Newtonian kinematics: displacement is calculated as s = v·dt + ½a·dt², then velocities are updated with v = v + a·dt. The simulation runs in a continuous loop, plotting each body’s incremental displacement with PLOT/DRAW to trace orbital paths on screen, with dt fixed at 0.1 time units per step.

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Program Structure

The program is organized into three logical phases separated by REM statements:

1. Input and initialization (lines 30–110): Collects parameters for both bodies, converts angles to radians, offsets x-coordinates by 10, draws initial circles, and decomposes velocities into x/y components.
2. Physics processing (lines 120–230): Computes gravitational force, resolves it into components, calculates accelerations, then updates displacements and velocities for both bodies.
3. Plotting and position update (lines 240–280): Draws each body’s step displacement on screen, updates coordinates, and loops back to line 120 unconditionally.

Physics Implementation

The simulation uses a fixed time step of dt = 0.1 (line 80). Newton’s law of gravitation is applied at line 130:

fg = g * m1 * m2 / ((x2-x1)² + (y2-y1)²)

This correctly computes the scalar gravitational force magnitude. The angle phi of the line joining the two bodies is found via ATN at line 150, with a special case at line 140 to avoid division by zero when the bodies share the same x-coordinate. Force components are then computed using COS phi and SIN phi, with SGN used to correctly assign direction based on relative position. Each body’s acceleration, displacement, and velocity are updated independently for x and y axes (lines 180–230) using the kinematic equation s = v·dt + ½·a·dt².

Notable Techniques

• Angle inputs are entered in degrees and converted to radians with * PI/180 at line 80, a common idiom.
• SGN (x2-x1) and SGN (y2-y1) at lines 160–170 elegantly handle the four quadrant cases for force direction without branching.
• The x-coordinates are shifted by +10 at line 80 (x1=x1+10, x2=x2+10), presumably to move bodies away from the left edge of the display area.
• PLOT/DRAW is used to trace orbital paths: PLOT x1,y1: DRAW s1x,s1y draws a line from the current position by the incremental displacement, effectively painting the trajectory incrementally.
• Initial body positions are marked with CIRCLE x1,y1,2 at line 90, providing a visual reference for starting locations.

Bugs and Anomalies

• Singularity on collision: Line 130 divides by the squared distance between the bodies. If the two bodies occupy the same position, this produces a division-by-zero error. There is no collision detection or minimum-distance guard.
• Force magnitude formula: The denominator at line 130 uses the squared distance, which is correct for the force magnitude. However, the angle phi is derived from ATN ABS(...) and then signed via SGN. Because ATN only returns values in (−π/2, π/2), taking the absolute value of the argument and reapplying sign via SGN is a valid workaround, but only works correctly because the force direction is reconstructed separately via SGN — the approach is correct but unconventional.
• Screen boundary: No bounds checking is performed on x or y coordinates. Bodies that drift off-screen will cause a PLOT/DRAW error.
• Euler integration drift: The integrator is first-order Euler for velocities (line 200/230) combined with a displacement formula that includes a second-order correction term. This mixed scheme is not fully second-order accurate (a proper Leapfrog or Verlet integrator would be more energy-conserving), so orbits will gradually drift over long runs.
• Variable name conflict: Input angles are stored as a1 and a2, which are then overwritten with radian-converted values at line 80. Later, a1x, a1y, a2x, a2y are used for accelerations (lines 180, 210). This is not a bug, but the reuse of a1/a2 for angle then implicitly dropped could confuse readers.

Key Variables

Variable	Description
m1, m2	Masses of the two bodies
v1, v2	Initial speed magnitudes
a1, a2	Launch angles (degrees, converted to radians in-place)
x1, y1, x2, y2	Current body coordinates
g	User-supplied gravitational constant
dt	Fixed time step (0.1)
fg	Scalar gravitational force between bodies
phi	Angle of the vector joining the two bodies
v1x, v1y, v2x, v2y	Velocity components
s1x, s1y, s2x, s2y	Displacement increments per time step