Mixing
Objects moving on a plane
$begingroup$
Object 1 starting at point (0,0) on the coordinate plane moves to right at 1 unit per second. Another object (object 2) starts at (x,y), moving half the distance to object 1 each second (following a curved path). If Object 2 is within 1 unit of Object 1, then they collide.
(1) Prove that they collide if and only if $x≥0$
(2) Find (and prove) what the maximum value of $y/x+y$ is if they do indeed collide.
I tried to solve (1) so far, without much luck. I tried powers of 2 and starting object 2 on the y axis. I got a nasty equation, and that was just for one point.
Object 1 moves first, so the movement of object 2 is half the distance to the position of object 1 at the end of the second, but the movement of object 2 is considered instantaneous
algebra-precalculus functions graphing-functions algorithmic-game-theory
asked Jan 22 at 0:23
mathboy1296mathboy1296
235
$endgroup$
add a comment |
$begingroup$
Object 1 starting at point (0,0) on the coordinate plane moves to right at 1 unit per second. Another object (object 2) starts at (x,y), moving half the distance to object 1 each second (following a curved path). If Object 2 is within 1 unit of Object 1, then they collide.
(1) Prove that they collide if and only if $x≥0$
(2) Find (and prove) what the maximum value of $y/x+y$ is if they do indeed collide.
I tried to solve (1) so far, without much luck. I tried powers of 2 and starting object 2 on the y axis. I got a nasty equation, and that was just for one point.
Object 1 moves first, so the movement of object 2 is half the distance to the position of object 1 at the end of the second, but the movement of object 2 is considered instantaneous
algebra-precalculus functions graphing-functions algorithmic-game-theory
asked Jan 22 at 0:23
mathboy1296mathboy1296
235
$endgroup$
add a comment |
$begingroup$
Object 1 starting at point (0,0) on the coordinate plane moves to right at 1 unit per second. Another object (object 2) starts at (x,y), moving half the distance to object 1 each second (following a curved path). If Object 2 is within 1 unit of Object 1, then they collide.
(1) Prove that they collide if and only if $x≥0$
(2) Find (and prove) what the maximum value of $y/x+y$ is if they do indeed collide.
I tried to solve (1) so far, without much luck. I tried powers of 2 and starting object 2 on the y axis. I got a nasty equation, and that was just for one point.
Object 1 moves first, so the movement of object 2 is half the distance to the position of object 1 at the end of the second, but the movement of object 2 is considered instantaneous
algebra-precalculus functions graphing-functions algorithmic-game-theory
asked Jan 22 at 0:23
mathboy1296mathboy1296
235
$endgroup$
Object 1 starting at point (0,0) on the coordinate plane moves to right at 1 unit per second. Another object (object 2) starts at (x,y), moving half the distance to object 1 each second (following a curved path). If Object 2 is within 1 unit of Object 1, then they collide.
(1) Prove that they collide if and only if $x≥0$
(2) Find (and prove) what the maximum value of $y/x+y$ is if they do indeed collide.
I tried to solve (1) so far, without much luck. I tried powers of 2 and starting object 2 on the y axis. I got a nasty equation, and that was just for one point.
Object 1 moves first, so the movement of object 2 is half the distance to the position of object 1 at the end of the second, but the movement of object 2 is considered instantaneous
algebra-precalculus functions graphing-functions algorithmic-game-theory
algebra-precalculus functions graphing-functions algorithmic-game-theory
asked Jan 22 at 0:23
mathboy1296mathboy1296
235
asked Jan 22 at 0:23
mathboy1296mathboy1296
235
asked Jan 22 at 0:23
mathboy1296mathboy1296
235
asked Jan 22 at 0:23
mathboy1296mathboy1296
235
asked Jan 22 at 0:23
mathboy1296mathboy1296
235
235
add a comment |
add a comment |
1 Answer
1
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$begingroup$
I'm afraid that (1) is not true.
Assume it is true. Then if $x<0$ the object never collide.
Consider now the situation $x=0$, $y=1000$. the objects will collide. After one second object 2 is a position $(0.5,500)$ and object one at $(1,0)$. Which is exactly the same situation as $x=-0.5$ and $y=500$ hence the object will not collide. Contradiction.
answered Jan 23 at 10:33
wecewece
2,3721923
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1 Answer
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1 Answer
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$begingroup$
I'm afraid that (1) is not true.
Assume it is true. Then if $x<0$ the object never collide.
Consider now the situation $x=0$, $y=1000$. the objects will collide. After one second object 2 is a position $(0.5,500)$ and object one at $(1,0)$. Which is exactly the same situation as $x=-0.5$ and $y=500$ hence the object will not collide. Contradiction.
answered Jan 23 at 10:33
wecewece
2,3721923
$endgroup$
add a comment |
$begingroup$
I'm afraid that (1) is not true.
Assume it is true. Then if $x<0$ the object never collide.
Consider now the situation $x=0$, $y=1000$. the objects will collide. After one second object 2 is a position $(0.5,500)$ and object one at $(1,0)$. Which is exactly the same situation as $x=-0.5$ and $y=500$ hence the object will not collide. Contradiction.
answered Jan 23 at 10:33
wecewece
2,3721923
$endgroup$
add a comment |
$begingroup$
I'm afraid that (1) is not true.
Assume it is true. Then if $x<0$ the object never collide.
Consider now the situation $x=0$, $y=1000$. the objects will collide. After one second object 2 is a position $(0.5,500)$ and object one at $(1,0)$. Which is exactly the same situation as $x=-0.5$ and $y=500$ hence the object will not collide. Contradiction.
answered Jan 23 at 10:33
wecewece
2,3721923
$endgroup$
I'm afraid that (1) is not true.
Assume it is true. Then if $x<0$ the object never collide.
Consider now the situation $x=0$, $y=1000$. the objects will collide. After one second object 2 is a position $(0.5,500)$ and object one at $(1,0)$. Which is exactly the same situation as $x=-0.5$ and $y=500$ hence the object will not collide. Contradiction.
answered Jan 23 at 10:33
wecewece
2,3721923
answered Jan 23 at 10:33
wecewece
2,3721923
answered Jan 23 at 10:33
wecewece
2,3721923
answered Jan 23 at 10:33
wecewece
2,3721923
2,3721923
add a comment |
add a comment |
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