Uphill Shooting (99)

Short description of my problem? I seem to get distances that are about 25-50% too long each time. For example, with the example input...

Accepted answer: 96 68 120 60 88 120 97 68 76

My answer: 159 91 152 82 115 147 148 108 94

I'm using object oriented programming and numeric integration to solve the problem. I assume projectiles start at the origin, fly in quadrant I and positive vectors are to the right and up (e.g. d2y = -9.8).

My projectile class has an x-component and y-component for three dimensions (e.g. x = 0, dx = v * cos a, d2x = 0 at initialization). My pseudo code is this:

My projectile.update(dt) function takes an amount of time to integrate across, then does the following:

1. Update time (t += dt)
2. Update position (x += dx * dt)
3. Update velocity (dx += d2x * dt)
4. (Acceleration is not updated)
5. Update projectile 2D speed and angle (not required by problem, for debugging)

My main function has the following pseudocode:

For each elevation profile of 3 profiles:
    Read the profile from standard input
    Transform profile from a list of n 4 meter blocks to a list of 4n 1 meter blocks
    For each shot of 3 shots:
        Read the projectile's velocity and angle from standard input
        Create an instance of Projectile class for that shot

        While the shot is at t=0 or its y > elevation[floor(x)]:
            If the shot is within one integration unit of impact:
                dt /= 10 //get a more accurate measurement of impact time and location
            Update the projectile for dt time

        Add the impact x value to the list of impacts

Print the impacts as required by the problem statement

Am I doing this right? Or is there a better way to do this and calculate the impact directly without numeric integration?

For purposes of troubleshooting, here is some data.

The results of transforming the elevation profile (slope) is: From: 0 0 0 0 0 1 1 1 1 2 2 2 2 2 2 3 4 5 6 6 7 8 9 9 10 10 10 10 10 10 11 11 12 13 14 15 16 16 17 18 To: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 13 13 13 13 14 14 14 14 15 15 15 15 16 16 16 16 16 16 16 16 17 17 17 17 18 18 18 18

I think my constructor function is correct. For the first shot, v = 42, a = 42, I get:

t: 0.000 x: 0.00 y: 0.00 dx: 31.21 dy: 28.10 d2x: 0.00 d2y: -9.81 vel: 42 ang: 42 elevation: 0

I also think my update function is correct. For dt = 0.1, after one integration, I get:

t: 0.100 x: 3.12 y: 2.81 dx: 31.21 dy: 27.12 d2x: 0.00 d2y: -9.81 vel: 41 ang: 40 elevation: 0

At impact for the first profile, first shot I have the following data: t: 5.111 x: 159.52 y: 18.01 dx: 31.21 dy: -22.04 d2x: 0.00 d2y: -9.81 vel: 38 ang: -35 elevation: 18 t: 5.112 x: 159.56 y: 17.99 dx: 31.21 dy: -22.05 d2x: 0.00 d2y: -9.81 vel: 38 ang: -35 elevation: 18

That was part of it. I updated the part where I parse the slope into a meter-by-meter elevations list. Now the elevations are in 4 meter steps.

Still doesn't solve the problem entirely. It brings me very close. Here's what I got after fixing the elevations problem: Expected output: 96 68 120 60 88 120 97 68 76 My output: 108 68 120 60 89 120 97 68 76

I had a similar outcome when I used the actual problem data. Matches on most, very close on the rest. Input:

0 0 0 0 0 0 0 1 2 2 2 2 2 2 3 4 4 4 4 5 5 6 7 8 8 9 10 10 10 10 11 11 11 12 13 13 14 15 15 16
30 32
42 44
32 49
0 0 0 0 0 0 0 1 2 2 2 3 3 4 5 5 6 7 7 8 8 9 10 11 11 11 12 12 12 13 13 13 13 13 13 14 15 16 17 17
36 67
45 60
43 62
0 0 0 0 0 1 1 2 2 3 3 4 5 5 5 6 7 7 7 8 8 9 10 11 12 13 14 15 15 15 15 15 15 15 15 16 16 17 17 17
39 51
34 41
39 45

Expected output:

56 114 81 78 140 120 94 64 88

My output: 58 120 84 78 140 120 94 64 88

I remember on "Safe Landing", we got a bit of a 'slop' factor in acceptable ranges of answers. Why do we have to match exactly here?

So I tried using your direct calculation function of y(x) = y0 + x*tan(a) - (g*x*x)/2*(v*cos(a))^2. It seems to have fixed the first case, but let's look at slope 0, shot 1 (v=42, a=44).

The accepted answer is impact at x=68. But here's the results of the direct calculation:

x= 67, y = 16.430351241442786, slope elevation = 16.0
x= 68, y = 16.26536937977803, slope elevation = 20.0

So I think that the answer is 67, no? It looks like the projectile hits the side of the step. Same thing happens on slope 0, shot 2. The accepted answer is x = 120. Here are the results of the calculation:

x = 119, y = 43.29930816948277 , slope elevation = 40.0
x = 120, y = 42.73938783509884, slope elevation = 44.0

Anyways, for the sample data, I now return the result 96 67 119 59 88 120 97 67 76

Thanks for the continued help, Quandray.