Fix bugs in cannon sim and tutorial.

docs/tutorial/ATutASimpleSim.md

@@ -76,7 +76,11 @@ y-coordinate again reaches 0.
 
 Solving for t (using the quadratic formula), we get the time of impact:
 
-![equation_analytic_t_impact](images/equation_analytic_t_impact.png)
+```math
+           -v_y0 - sqrt(v_y0^2 - 2*g*y0)
+t_impact = -----------------------------
+                         g
+```
 
 ---
 <a id=a-cannonball-simulation-without-trick></a>
@@ -88,57 +92,59 @@ Solving for t (using the quadratic formula), we get the time of impact:
 ```c
 /* Cannonball without Trick */
 
-#include <stdio.h>
 #include <math.h>
+#include <stdio.h>
 
-int main (int argc, char * argv[]) {
-
-    /* Declare variables used in the simulation */
-    double pos[2]; double pos_orig[2] ;
-    double vel[2]; double vel_orig[2] ;
-    double acc[2];
-    double init_angle ;
-    double init_speed ;
-    double time ;
-    int impact;
-    double impactTime;
-
-    /* Initialize data */
-    pos[0] = 0.0 ; pos[1] = 0.0 ;
-    vel[0] = 0.0 ; vel[1] = 0.0 ;
-    acc[0] = 0.0 ; acc[1] = -9.81 ;
-    time = 0.0 ;
-    init_angle = M_PI/6.0 ;
-    init_speed = 50.0 ;
-    impact = 0;
-
-    /* Do initial calculations */
-    pos_orig[0] = pos[0] ;
-    pos_orig[1] = pos[1] ;
-    vel_orig[0] = cos(init_angle)*init_speed ;
-    vel_orig[1] = sin(init_angle)*init_speed ;
+int main(void)
+{
+    /* Initial conditions */
+    const double acc[2]     = {0.0, -9.81}; // acceleration in m/s^2
+    const double init_angle = M_PI / 6.0;   // initial angle in radians
+    const double init_speed = 50.0;         // initial speed in m/s
+    const double time_step  = 0.01;         // time step in seconds
+
+    const double init_pos[2] = {0.0, 0.0};  // initial position in meters
+    const double init_vel[2]                // initial velocity in m/s
+        = {
+            cos(init_angle) * init_speed,
+            sin(init_angle) * init_speed,
+        };
+
+    /* Initialize simulation state */
+    double pos[2]      = {init_pos[0], init_pos[1]}; // current position in meters
+    double vel[2]      = {init_vel[0], init_vel[1]}; // current velocity in m/s
+    double sim_time    = 0.0;                        // current simulation time in seconds
+    double impact_time = 0.0;                        // time of impact in seconds
+    int impact         = 0; // flag indicating whether an impact has occurred
+
+    printf("time, pos[0], pos[1], vel[0], vel[1]\n");
 
     /* Run simulation */
-    printf("time, pos[0], pos[1], vel[0], vel[1]\n" );
-    while ( !impact ) {
-        vel[0] = vel_orig[0] + acc[0] * time ;
-        vel[1] = vel_orig[1] + acc[1] * time ;
-        pos[0] = pos_orig[0] + (vel_orig[0] + 0.5 * acc[0] * time) * time ;
-        pos[1] = pos_orig[1] + (vel_orig[1] + 0.5 * acc[1] * time) * time ;
-        printf("%7.2f, %10.6f, %10.6f, %10.6f, %10.6f\n", time, pos[0], pos[1], vel[0], vel[1] );
-        if (pos[1] < 0.0) {
-            impact = 1;
-            impactTime = (- vel_orig[1] - 
-                          sqrt(vel_orig[1] * vel_orig[1] - 2.0 * pos_orig[1])
-                         ) / -9.81; 
-            pos[0] = impactTime * vel_orig[0];
+    while (!impact) {
+        vel[0] = init_vel[0] + acc[0] * sim_time;
+        vel[1] = init_vel[1] + acc[1] * sim_time;
+
+        pos[0] = init_pos[0] + (init_vel[0] + 0.5 * acc[0] * sim_time) * sim_time;
+        pos[1] = init_pos[1] + (init_vel[1] + 0.5 * acc[1] * sim_time) * sim_time;
+
+        printf("%7.2f, %10.6f, %10.6f, %10.6f, %10.6f\n", sim_time, pos[0], pos[1], vel[0], vel[1]);
+
+        if (pos[1] < 0.0) { // check for impact
+            impact_time
+                = (-init_vel[1] - sqrt(init_vel[1] * init_vel[1] - 2.0 * acc[1] * init_pos[1]))
+                / acc[1];
+
+            pos[0] = init_pos[0] + (init_vel[0] + 0.5 * acc[0] * impact_time) * impact_time;
             pos[1] = 0.0;
+
+            impact = 1;
         }
-        time += 0.01 ;
+
+        sim_time += time_step;
     }
 
     /* Shutdown simulation */
-        printf("Impact time=%lf position=%lf\n", impactTime, pos[0]);
+    printf("Impact time=%f position=%f\n", impact_time, pos[0]);
 
     return 0;
 }
@@ -147,7 +153,7 @@ int main (int argc, char * argv[]) {
 If we compile and run the program in listing 1:
 
 ```bash
-% cc cannon.c -o cannon -lm
+% cc cannon.c -o cannon
 % ./cannon
 ```