traj.out

pqR version 2.15.1 (2020-07-23), based on R 2.15.0 (2012-03-30)

R 2.15.0 is Copyright (C) 2012 The R Foundation for Statistical Computing
ISBN 3-900051-07-0

Modifications to R in pqR are Copyright (C) 2013-2020 Radford M. Neal

Some modules are from R-2.15.1 or later versions distributed by the R Core Team

Platform: x86_64-apple-darwin17.7.0 (64-bit)

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No helper threads, task merging enabled, uncompressed pointers.

> # Find the distance travelled by a projectile launched on level
> # ground with initial velocity (vx,vy), with no air resistance.
> 
> distance_travelled <- function (vx, vy, dt=0.0001, g=9.8) {
+     x <- y <- 0
+     repeat {
+         last_x <- x
+         last_y <- y
+         x <- x + vx*dt
+         y <- y + vy*dt
+         if (y < 0)  # return impact x location, interpolating
+             return ((x*last_y - last_x*y) / (last_y-y))
+         vy <- vy - g*dt
+     }
+ }
> 
> 
> # See the derivative of distance with respect to angle of launch for 
> # angles of 40 and 60 degrees.
> 
> with gradient (a = 40 * pi/180) distance_travelled (cos(a), sin(a))
[1] 0.1005672
attr(,"gradient")
[1] 0.03543097
> with gradient (a = 60 * pi/180) distance_travelled (cos(a), sin(a))
[1] 0.08841993
attr(,"gradient")
[1] -0.1021333
> 
> 
> # Find angle of launch maximizing distance, for fixed initial speed.
> # Note that nlm minimizes, so we negate the distance.
> 
> system.time(print(
+ nlm (function (a) 
+        with gradient (a) -distance_travelled (cos(a), sin(a)),
+      0) $ estimate * 180/pi
+ ))
[1] 44.99701
   user  system elapsed 
  0.099   0.007   0.106 
> 
> 
> # For this simple example with only one variable, optimizing without
> # the gradient information is about as fast...
> 
> system.time(print(
+ nlm (function (a) 
+        -distance_travelled (cos(a), sin(a)),
+      0) $ estimate * 180/pi
+ ))
[1] 44.99701
   user  system elapsed 
  0.071   0.000   0.072 
>