A simulated solution for rabbit hunting

A rabbit is at, and its cave is at, 20 m north. A wolf is at, 33 m east of the rabbit: the wolf stared at the rabbit as quickly as the rabbit and asked if the rabbit would be caught before arriving at the hole?

Many people on the Internet say they can be caught up, including a similar question, which is 100 yards away from the rabbit and the hole is 60 yards away. So can we?

Through computer simulation, you can find the answer without solving the differential equation.

To simulate a computer, it basically uses the push-down algorithm to give the initial conditions and calculate the next step based on the formula until the calculation is terminated under certain conditions.

When we look at the problem of simulated Chase, we can look at the time slice to see what the next step is, and here we can convert the speed to the stepping distance.

In the plane coordinate system, the rabbit is located at the origin O, and the positive north is the y-axis positive. The wolf is traced from the I quadrant.

 

[Cpp] # include <stdio. h>
# Include <math. h>
 
Double Td, Tx, Ty, Ld, Lx, Ly;
 
Double dist ()
{
Return sqrt (Tx-Lx) * (Tx-Lx) + (Ty-Ly) * (Ty-Ly ));
}
 
// The Wolf's position changes and points to the rabbit's line segment to run the Ld distance, corresponding to the ldx and ldy Components
Double NextL ()
{
Double dx = fabs (Tx-Lx );
Double dy = fabs (Ty-Ly );
Double k, dk;
Int usek = 1;
Double ldx;
Double ldy;
//
If (dx> 0.00000000001)
{
K = dy/dx;
} Else
{
Usek = 0;
Dk = dx/dy;
}
//
If (usek = 1)
{
Ldx = sqrt (Ld * Ld)/(k * k + 1 ));
Ldy = k * ldx;
} Else
{
Ldy = sqrt (Ld * Ld)/(dk * dk + 1 ));
Ldx = dk * ldy;
}
// I quadrant only
Lx = Lx-ldx;
Ly = Ly + ldy;
}
 
// Rabbit location change
Double NextT ()
{
Tx = Tx;
Ty = Ty + Td;
}
 
Void loop ()
{
// Coordinate creation and initialization
Tx = 0.0;
Ty = 0.0;
Lx = 33.0;
Ly = 0.0;
// Rabbit walking distance
Td = 0.1;
// Wolf stepping distance
Ld = Td * 2.0;
While (dist ()> = Td)
{
NextL ();
NextT ();
Printf ("Rabbit :( % lf, % lf) Wolf :( % lf, % lf) \ n", Tx, Ty, Lx, Ly );
}
//
}
 
Int main (int argc, char ** argv)
{
Loop ();
Return 0;
}

# Include <stdio. h>
# Include <math. h>

Double Td, Tx, Ty, Ld, Lx, Ly;

Double dist ()
{
Return sqrt (Tx-Lx) * (Tx-Lx) + (Ty-Ly) * (Ty-Ly ));
}

// The Wolf's position changes and points to the rabbit's line segment to run the Ld distance, corresponding to the ldx and ldy Components
Double NextL ()
{
Double dx = fabs (Tx-Lx );
Double dy = fabs (Ty-Ly );
Double k, dk;
Int usek = 1;
Double ldx;
Double ldy;
//
If (dx> 0.00000000001)
{
K = dy/dx;
} Else
{
Usek = 0;
Dk = dx/dy;
}
//
If (usek = 1)
{
Ldx = sqrt (Ld * Ld)/(k * k + 1 ));
Ldy = k * ldx;
} Else
{
Ldy = sqrt (Ld * Ld)/(dk * dk + 1 ));
Ldx = dk * ldy;
}
// I quadrant only
Lx = Lx-ldx;
Ly = Ly + ldy;
}

// Rabbit location change
Double NextT ()
{
Tx = Tx;
Ty = Ty + Td;
}

Void loop ()
{
// Coordinate creation and initialization
Tx = 0.0;
Ty = 0.0;
Lx = 33.0;
Ly = 0.0;
// Rabbit walking distance
Td = 0.1;
// Wolf stepping distance
Ld = Td * 2.0;
While (dist ()> = Td)
{
NextL ();
NextT ();
Printf ("Rabbit :( % lf, % lf) Wolf :( % lf, % lf) \ n", Tx, Ty, Lx, Ly );
}
//
}

Int main (int argc, char ** argv)
{
Loop ();
Return 0;
}

Here, NextT provides the formula for pushing the rabbit's position, and NextL provides the formula for pushing the wolf's position, in the cycle, because the wolf uses the rabbit's seat to calculate the next step, he must shift the rabbit's position (or vice versa, that is, the wolf can slightly predict the rabbit's movements, if a wolf can predict whether a rabbit's path is completely straight, he can catch the rabbit. However, the fact is usually a lot of chasing .)

Running result

Rabbit :( 0.000000, 0.100000) Wolf :( 32.800000, 0.000000)
Rabbit :( 0.000000, 0.200000) Wolf :( 32.600001, 0.000610)
Rabbit :( 0.000000, 0.300000) Wolf :( 32.400005, 0.001833)
Rabbit :( 0.000000, 0.400000) Wolf :( 32.200013, 0.003673)
Rabbit :( 0.000000, 0.500000) Wolf :( 32.000028, 0.006135)
Rabbit :( 0.000000, 0.600000) Wolf :( 31.800052, 0.009221)
Rabbit :( 0.000000, 0.700000) Wolf :( 31.600087, 0.012936)
Rabbit :( 0.000000, 0.800000) Wolf :( 31.400134, 0.017284)
Rabbit :( 0.000000, 0.900000) Wolf :( 31.200196, 0.022267)
Rabbit :( 0.000000, 1.000000) Wolf :( 31.000275, 0.027892)
Rabbit :( 0.000000, 1.100000) Wolf :( 30.800373, 0.034160)
Rabbit :( 0.000000, 1.200000) Wolf :( 30.600493, 0.041077)
Rabbit :( 0.000000, 1.300000) Wolf :( 30.400636, 0.048646)
Rabbit :( 0.000000, 1.400000) Wolf :( 30.200805, 0.056872)
Rabbit :( 0.000000, 1.500000) Wolf :( 30.001003, 0.065757)
Rabbit :( 0.000000, 1.600000) Wolf :( 29.801231, 0.075308)
Rabbit :( 0.000000, 1.700000) Wolf :( 29.601492, 0.085527)
Rabbit :( 0.000000, 1.800000) Wolf :( 29.401789, 0.096419)
Rabbit :( 0.000000, 1.900000) Wolf :( 29.202124, 0.107988)
Rabbit :( 0.000000, 2.000000) Wolf :( 29.002500, 0.120238)
Rabbit :( 0.000000, 2.100000) Wolf :( 28.802918, 0.133173)
Rabbit :( 0.000000, 2.200000) Wolf :( 28.603383, 0.146799)
Rabbit :( 0.000000, 2.300000) Wolf :( 28.403896, 0.161118)
Rabbit :( 0.000000, 2.400000) Wolf :( 28.204461, 0.176136)
Rabbit :( 0.000000, 2.500000) Wolf :( 28.005080, 0.191857)
Rabbit :( 0.000000, 2.600000) Wolf :( 27.805756, 0.208285)
Rabbit :( 0.000000, 2.700000) Wolf :( 27.606491, 0.225425)
Rabbit :( 0.000000, 2.800000) Wolf :( 27.407290, 0.243281)
Rabbit :( 0.000000, 2.900000) Wolf :( 27.208155, 0.261857)
Rabbit :( 0.000000, 3.000000) Wolf :( 27.009088, 0.281159)
Rabbit :( 0.000000, 3.100000) Wolf :( 26.810094, 0.301191)
Rabbit :( 0.000000, 3.200000) Wolf :( 26.611175, 0.321957)
Rabbit :( 0.000000, 3.300000) Wolf :( 26.412334, 0.343462)
Rabbit :( 0.000000, 3.400000) Wolf :( 26.213576, 0.365710)
Rabbit :( 0.000000, 3.500000) Wolf :( 26.014902, 0.388707)
Rabbit :( 0.000000, 3.600000) Wolf :( 25.816317, 0.412457)
Rabbit :( 0.000000, 3.700000) Wolf :( 25.617825, 0.436965)
Rabbit :( 0.000000, 3.800000) Wolf :( 25.419428, 0.462236)
Rabbit :( 0.000000, 3.900000) Wolf :( 25.221130, 0.488274)
Rabbit :( 0.000000, 4.000000) Wolf :( 25.022935, 0.515084)
Rabbit :( 0.000000, 4.100000) Wolf :( 24.824847, 0.542671)
Rabbit :( 0.000000, 4.200000) Wolf :( 24.626869, 0.571041)
Rabbit :( 0.000000, 4.300000) Wolf :( 24.429006, 0.600198)
Rabbit :( 0.000000, 4.400000) Wolf :( 24.231261, 0.630146)
Rabbit :( 0.000000, 4.500000) Wolf :( 24.033638, 0.660892)
Rabbit :( 0.000000, 4.600000) Wolf :( 23.836142, 0.692440)
Rabbit :( 0.000000, 4.700000) Wolf :( 23.638776, 0.724795)
Rabbit :( 0.000000, 4.800000) Wolf :( 23.441546, 0.757962)
Rabbit :( 0.000000, 4.900000) Wolf :( 23.244454, 0.791947)
Rabbit :( 0.000000, 5.000000) Wolf :( 23.047506, 0.826754)
Rabbit :( 0.000000, 5.100000) Wolf :( 22.850707, 0.862389)
Rabbit :( 0.000000, 5.200000) Wolf :( 22.654060, 0.898857)
Rabbit :( 0.000000, 5.300000) Wolf :( 22.457570, 0.936162)
Rabbit :( 0.000000, 5.400000) Wolf :( 22.261242, 0.974312)
Rabbit :( 0.000000, 5.500000) Wolf :( 22.065081, 1.013310)
Rabbit :( 0.000000, 5.600000) Wolf :( 21.869092, 1.053162)
Rabbit :( 0.000000, 5.700000) Wolf :( 21.673279, 1.093874)
Rabbit :( 0.000000, 5.800000) Wolf :( 21.477648, 1.135451)
Rabbit :( 0.000000, 5.900000) Wolf :( 21.282204, 1.177897)
Rabbit :( 0.000000, 6.000000) Wolf :( 21.086953, 1.221220)
Rabbit :( 0.000000, 6.100000) Wolf :( 20.891899, 1.265423)
Rabbit :( 0.000000, 6.200000) Wolf :( 20.697048, 1.310514)
Rabbit :( 0.000000, 6.300000) Wolf :( 20.502406, 1.356496)
Rabbit :( 0.000000, 6.400000) Wolf :( 20.307978, 1.403376)
Rabbit :( 0.000000, 6.500000) Wolf :( 20.113770, 1.451160)
Rabbit :( 0.000000, 6.600000) Wolf :( 19.919788, 1.499852)
Rabbit :( 0.000000, 6.700000) Wolf :( 19.726037, 1.549459)
Rabbit :( 0.000000, 6.800000) Wolf :( 19.532525, 1.599985)
Rabbit :( 0.000000, 6.900000) Wolf :( 19.339257, 1.651438)
Rabbit :( 0.000000, 7.000000) Wolf :( 19.146239, 1.703822)
Rabbit :( 0.000000, 7.100000) Wolf :( 18.953478, 1.757143)
Rabbit :( 0.000000, 7.200000) Wolf :( 18.760980, 1.811407)
Rabbit :( 0.000000, 7.300000) Wolf :( 18.568752, 1.866619)
Rabbit :( 0.000000, 7.400000) Wolf :( 18.376800, 1.922786)
Rabbit :( 0.000000, 7.500000) Wolf :( 18.185133, 1.979913)
Rabbit :( 0.000000, 7.600000) Wolf :( 17.993755, 2.038005)
Rabbit :( 0.000000, 7.700000) Wolf :( 17.802676, 2.097069)
Rabbit :( 0.000000, 7.800000) Wolf :( 17.611901, 2.157110)
Rabbit :( 0.000000, 7.900000) Wolf :( 17.421438, 2.218135)
Rabbit :( 0.000000, 8.000000) Wolf :( 17.231295, 2.280149)
Rabbit :( 0.000000, 8.100000) Wolf :( 17.041480, 2.343157)
Rabbit :( 0.000000, 8.200000) Wolf :( 16.851999, 2.407166)
Rabbit :( 0.000000, 8.300000) Wolf :( 16.662862, 2.472182)
Rabbit :( 0.000000, 8.400000) Wolf :( 16.474076, 2.538210)
Rabbit :( 0.000000, 8.500000) Wolf :( 16.285648, 2.605256)
Rabbit :( 0.000000, 8.600000) Wolf :( 16.097589, 2.673326)
Rabbit :( 0.000000, 8.700000) Wolf :( 15.909905, 2.742425)
Rabbit :( 0.000000, 8.800000) Wolf :( 15.722605, 2.812561)
Rabbit :( 0.000000, 8.900000) Wolf :( 15.535700, 2.883738)
Rabbit :( 0.000000, 9.000000) Wolf :( 15.349196, 2.955962)
Rabbit :( 0.000000, 9.100000) Wolf :( 15.163103, 3.029240)
Rabbit :( 0.000000, 9.200000) Wolf :( 14.977431, 3.103576)
Rabbit :( 0.000000, 9.300000) Wolf :( 14.792189, 3.178977)
Rabbit :( 0.000000, 9.400000) Wolf :( 14.607386, 3.255449)
Rabbit :( 0.000000, 9.500000) Wolf :( 14.423032, 3.332997)
Rabbit :( 0.000000, 9.600000) Wolf :( 14.239137, 3.411627)
Rabbit :( 0.000000, 9.700000) Wolf :( 14.055711, 3.491344)
Rabbit :( 0.000000, 9.800000) Wolf :( 13.872764, 3.572155)
Rabbit :( 0.000000, 9.900000) Wolf :( 13.690307, 3.654065)
Rabbit :( 0.000000, 10.000000) Wolf :( 13.508349, 3.737080)
Rabbit :( 0.000000, 10.100000) Wolf :( 13.326902, 3.821204)
Rabbit :( 0.000000, 10.200000) Wolf :( 13.145977, 3.906445)
Rabbit :( 0.000000, 10.300000) Wolf :( 12.965584, 3.992807)
Rabbit :( 0.000000, 10.400000) Wolf :( 12.785735, 4.080296)
Rabbit :( 0.000000, 10.500000) Wolf :( 12.606441, 4.168917)
Rabbit :( 0.000000, 10.600000) Wolf :( 12.427714, 4.258675)
Rabbit :( 0.000000, 10.700000) Wolf :( 12.249565, 4.349577)
Rabbit :( 0.000000, 10.800000) Wolf :( 12.072007, 4.441627)
Rabbit :( 0.000000, 10.900000) Wolf :( 11.895052, 4.534830)
Rabbit :( 0.000000, 11.000000) Wolf :( 11.718711, 4.629191)
Rabbit :( 0.000000, 11.100000) Wolf :( 11.542999, 4.724716)
Rabbit :( 0.000000, 11.200000) Wolf :( 11.367926, 4.821410)
Rabbit :( 0.000000, 11.300000) Wolf :( 11.193507, 4.919277)
Rabbit :( 0.000000, 11.400000) Wolf :( 11.019755, 5.018323)
Rabbit :( 0.000000, 11.500000) Wolf :( 10.846682, 5.118552)
Rabbit :( 0.000000, 11.600000) Wolf :( 10.674302, 5.219968)
Rabbit :( 0.000000, 11.700000) Wolf :( 10.502630, 5.322577)
Rabbit :( 0.000000, 11.800000) Wolf :( 10.331678, 5.426382)
Rabbit :( 0.000000, 11.900000) Wolf :( 10.161462, 5.531389)
Rabbit :( 0.000000, 12.000000) Wolf :( 9.991995, 5.637601)
Rabbit :( 0.000000, 12.100000) Wolf :( 9.823292, 5.745022)
Rabbit :( 0.000000, 12.200000) Wolf :( 9.655368, 5.853657)
Rabbit :( 0.000000, 12.300000) Wolf :( 9.488238, 5.963509)
Rabbit :( 0.000000, 12.400000) Wolf :( 9.321917, 6.074583)
Rabbit :( 0.000000, 12.500000) Wolf :( 9.156421, 6.186881)
Rabbit :( 0.000000, 12.600000) Wolf :( 8.991764, 6.300408)
Rabbit :( 0.000000, 12.700000) Wolf :( 8.827964, 6.415166)
Rabbit :( 0.000000, 12.800000) Wolf :( 8.665035, 6.531158)
Rabbit :( 0.000000, 12.900000) Wolf :( 8.502995, 6.648389)
Rabbit :( 0.000000, 13.000000) Wolf :( 8.341859, 6.766859)
Rabbit :( 0.000000, 13.100000) Wolf :( 8.181645, 6.886573)
Rabbit :( 0.000000, 13.200000) Wolf :( 8.022369, 7.007533)
Rabbit :( 0.000000, 13.300000) Wolf :( 7.864049, 7.129740)
Rabbit :( 0.000000, 13.400000) Wolf :( 7.706701, 7.253198)
Rabbit :( 0.000000, 13.500000) Wolf :( 7.550344, 7.377907)
Rabbit :( 0.000000, 13.600000) Wolf :( 7.394995, 7.503870)
Rabbit :( 0.000000, 13.700000) Wolf :( 7.240672, 7.631088)
Rabbit :( 0.000000, 13.800000) Wolf :( 7.087393, 7.759561)
Rabbit :( 0.000000, 13.900000) Wolf :( 6.935176, 7.889292)
Rabbit :( 0.000000, 14.000000) Wolf :( 6.784041, 8.020281)
Rabbit :( 0.000000, 14.100000) Wolf :( 6.634005, 8.152529)
Rabbit :( 0.000000, 14.200000) Wolf :( 6.485088, 8.286034)
Rabbit :( 0.000000, 14.300000) Wolf :( 6.337310, 8.420799)
Rabbit :( 0.000000, 14.400000) Wolf :( 6.190688, 8.556821)
Rabbit :( 0.000000, 14.500000) Wolf :( 6.045244, 8.694101)
Rabbit :( 0.000000, 14.600000) Wolf :( 5.900995, 8.832638)
Rabbit :( 0.000000, 14.700000) Wolf :( 5.757964, 8.972431)
Rabbit :( 0.000000, 14.800000) Wolf :( 5.616169, 9.113478)
Rabbit :( 0.000000, 14.900000) Wolf :( 5.475630, 9.255776)
Rabbit :( 0.000000, 15.000000) Wolf :( 5.336369, 9.399326)
Rabbit :( 0.000000, 15.100000) Wolf :( 5.198406, 9.544122)
Rabbit :( 0.000000, 15.200000) Wolf :( 5.061761, 9.690164)
Rabbit :( 0.000000, 15.300000) Wolf :( 4.926455, 9.837447)
Rabbit :( 0.000000, 15.400000) Wolf :( 4.792509, 9.985968)
Rabbit :( 0.000000, 15.500000) Wolf :( 4.659945, 10.135724)
Rabbit :( 0.000000, 15.600000) Wolf :( 4.528784, 10.286710)
Rabbit :( 0.000000, 15.700000) Wolf :( 4.399047, 10.438921)
Rabbit :( 0.000000, 15.800000) Wolf :( 4.270755, 10.592353)
Rabbit :( 0.000000, 15.900000) Wolf :( 4.143931, 10.746999)
Rabbit :( 0.000000, 16.000000) Wolf :( 4.018595, 10.902855)
Rabbit :( 0.000000, 16.100000) Wolf :( 3.894770, 11.059913)
Rabbit :( 0.000000, 16.200000) Wolf :( 3.772477, 11.218168)
Rabbit :( 0.000000, 16.300000) Wolf :( 3.651739, 11.377612)
Rabbit :( 0.000000, 16.400000) Wolf :( 3.532577, 11.538237)
Rabbit :( 0.000000, 16.500000) Wolf :( 3.415014, 11.700035)
Rabbit :( 0.000000, 16.600000) Wolf :( 3.299070, 11.862999)
Rabbit :( 0.000000, 16.700000) Wolf :( 3.184770, 12.027119)
Rabbit :( 0.000000, 16.800000) Wolf :( 3.072133, 12.192386)
Rabbit :( 0.000000, 16.900000) Wolf :( 2.961183, 12.358789)
Rabbit :( 0.000000, 17.000000) Wolf :( 2.851942, 12.526319)
Rabbit :( 0.000000, 17.100000) Wolf :( 2.744432, 12.694965)
Rabbit :( 0.000000, 17.200000) Wolf :( 2.638673, 12.864716)
Rabbit :( 0.000000, 17.300000) Wolf :( 2.534690, 13.035559)
Rabbit :( 0.000000, 17.400000) Wolf :( 2.432502, 13.207482)
Rabbit :( 0.000000, 17.500000) Wolf :( 2.332132, 13.380473)
Rabbit :( 0.000000, 17.600000) Wolf :( 2.233602, 13.554519)
Rabbit :( 0.000000, 17.700000) Wolf :( 2.136933, 13.729605)
Rabbit :( 0.000000, 17.800000) Wolf :( 2.042147, 13.905717)
Rabbit :( 0.000000, 17.900000) Wolf :( 1.949264, 14.082841)
Rabbit :( 0.000000, 18.000000) Wolf :( 1.858306, 14.260960)
Rabbit :( 0.000000, 18.100000) Wolf :( 1.769293, 14.440060)
Rabbit :( 0.000000, 18.200000) Wolf :( 1.682246, 14.620124)
Rabbit :( 0.000000, 18.300000) Wolf :( 1.597186, 14.801134)
Rabbit :( 0.000000, 18.400000) Wolf :( 1.514133, 14.983074)
Rabbit :( 0.000000, 18.500000) Wolf :( 1.433107, 15.165926)
Rabbit :( 0.000000, 18.600000) Wolf :( 1.354126, 15.349671)
Rabbit :( 0.000000, 18.700000) Wolf :( 1.277212, 15.534289)
Rabbit :( 0.000000, 18.800000) Wolf :( 1.202382, 15.719763)
Rabbit :( 0.000000, 18.900000) Wolf :( 1.129656, 15.906072)
Rabbit :( 0.000000, 19.000000) Wolf :( 1.059052, 16.093195)
Rabbit :( 0.000000, 19.100000) Wolf :( 0.990587, 16.281111)
Rabbit :( 0.000000, 19.200000) Wolf :( 0.924280, 16.469800)
Rabbit :( 0.000000, 19.300000) Wolf :( 0.860147, 16.659239)
Rabbit :( 0.000000, 19.400000) Wolf :( 0.798206, 16.849405)
Rabbit :( 0.000000, 19.500000) Wolf :( 0.738473, 17.040277)
Rabbit :( 0.000000, 19.600000) Wolf :( 0.680964, 17.231830)
Rabbit :( 0.000000, 19.700000) Wolf :( 0.625694, 17.424041)
Rabbit :( 0.000000, 19.800000) Wolf :( 0.572678, 17.616887)
Rabbit :( 0.000000, 19.900000) Wolf :( 0.521931, 17.810341)
Rabbit :( 0.000000, 20.000000) Wolf :( 0.473466, 18.004380)
Rabbit :( 0.000000, 20.100000) Wolf :( 0.427297, 18.198979)
Rabbit :( 0.000000, 20.200000) Wolf :( 0.383437, 18.394110)
Rabbit :( 0.000000, 20.300000) Wolf :( 0.341898, 18.589749)
Rabbit :( 0.000000, 20.400000) Wolf :( 0.302691, 18.785868)
Rabbit :( 0.000000, 20.500000) Wolf :( 0.265829, 18.982442)
Rabbit :( 0.000000, 20.600000) Wolf :( 0.231320, 19.179442)
Rabbit :( 0.000000, 20.700000) Wolf :( 0.199176, 19.376842)
Rabbit :( 0.000000, 20.800000) Wolf :( 0.169405, 19.574614)
Rabbit :( 0.000000, 20.900000) Wolf :( 0.142017, 19.772730)
Rabbit :( 0.000000, 21.000000) Wolf :( 0.117018, 19.971161)
Rabbit :( 0.000000, 21.100000) Wolf :( 0.094416, 20.169880)
Rabbit :( 0.000000, 21.200000) Wolf :( 0.074218, 20.368858)
Rabbit :( 0.000000, 21.300000) Wolf :( 0.056429, 20.568065)
Rabbit :( 0.000000, 21.400000) Wolf :( 0.041056, 20.767473)
Rabbit :( 0.000000, 21.500000) Wolf :( 0.028101, 20.967053)
Rabbit :( 0.000000, 21.600000) Wolf :( 0.017570, 21.166776)
Rabbit :( 0.000000, 21.700000) Wolf :( 0.009466, 21.366611)
Rabbit :( 0.000000, 21.800000) Wolf :( 0.003789, 21.566531)
Rabbit :( 0.000000, 21.900000) Wolf :( 0.000544, 21.766505)

We can see that a wolf can catch up with a rabbit at 22 meters, but not at 20 meters.

When the distance is also 100 units, you must catch up with 66.6 units, but not 60 units.