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Number of bounces by an object in a unit square with velocity $vec(p,q)$, where p and q are co-prime.
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Finding beam paths using reflectionIs there solution for equation which is recursive?Find a function $f(x)$ so that the graph of $y=f(x)$ is the path of the particle.Finding magnitudes of tensionsFinding the distance between two moving objectsvectors: finding accelerationInitial Velocity Vector CalculationVector calculationsThe simplest billiards problemHard trajectory of equilateral triangle systemHow do you Graph the Trajectory of Position/Velocity Functions?
$begingroup$
Please bear with me.
Imagine the unit square in the plane to be a carrom board. Assume the striker is just
a point, moving with no friction (so it goes forever), and that when it hits an edge, the
angle of reflection is equal to the angle of incidence, as in real life. If the striker ever
hits a corner it falls into the pocket and disappears. The trajectory of the striker is
completely determined by its starting point (x, y) and its initial velocity $$vec(p, q)$$.
If the striker eventually returns to its initial state (i.e. initial position and initial velocity),
we define its bounce number to be the number of edges it hits before returning to its initial
state for the first time.
For example, the traectory with initial state $$[(.5, .5);vec(1, 0)]$$ has bounce number 2 and
it returns to its initial state for the first time in 2 time units. And the trajectory with
initial state $$[(.25, .75); vec(1, 1)] $$has bounce number 4.
(a) Suppose the striker has initial state $[(.5, .5);vec(p, q)]$. If p > q ≥ 0 then what is the
velocity after it hits an edge for the first time? What if q > p ≥ 0?
(b) Draw a trajectory with bounce number 5 or justify why it is impossible.
(c) Consider the trajectory with initial state $[(x, y); vec(p, 0)]$ where p is a positive integer.
In how much time will the striker first return to its initial state?
(d) What is the bounce number for the initial state $[(x, y);vec
(p, q)]$ where p, q are relatively
prime positive integers, assuming the striker never hits a corner?
I have done (a), (b) and (c) and am stuck on (d).
Answer to (a) is $vec(-p,q)$ and $vec(p,-q)$.
Answer to (c) is $$frac2p$$
Can you please tell me how to do it?
Thanks.
vectors
asked Apr 1 at 17:32
RaghavRaghav
627
$endgroup$
add a comment |
$begingroup$
Please bear with me.
Imagine the unit square in the plane to be a carrom board. Assume the striker is just
a point, moving with no friction (so it goes forever), and that when it hits an edge, the
angle of reflection is equal to the angle of incidence, as in real life. If the striker ever
hits a corner it falls into the pocket and disappears. The trajectory of the striker is
completely determined by its starting point (x, y) and its initial velocity $$vec(p, q)$$.
If the striker eventually returns to its initial state (i.e. initial position and initial velocity),
we define its bounce number to be the number of edges it hits before returning to its initial
state for the first time.
For example, the traectory with initial state $$[(.5, .5);vec(1, 0)]$$ has bounce number 2 and
it returns to its initial state for the first time in 2 time units. And the trajectory with
initial state $$[(.25, .75); vec(1, 1)] $$has bounce number 4.
(a) Suppose the striker has initial state $[(.5, .5);vec(p, q)]$. If p > q ≥ 0 then what is the
velocity after it hits an edge for the first time? What if q > p ≥ 0?
(b) Draw a trajectory with bounce number 5 or justify why it is impossible.
(c) Consider the trajectory with initial state $[(x, y); vec(p, 0)]$ where p is a positive integer.
In how much time will the striker first return to its initial state?
(d) What is the bounce number for the initial state $[(x, y);vec
(p, q)]$ where p, q are relatively
prime positive integers, assuming the striker never hits a corner?
I have done (a), (b) and (c) and am stuck on (d).
Answer to (a) is $vec(-p,q)$ and $vec(p,-q)$.
Answer to (c) is $$frac2p$$
Can you please tell me how to do it?
Thanks.
vectors
asked Apr 1 at 17:32
RaghavRaghav
627
$endgroup$
$begingroup$
See math.stackexchange.com/questions/2800897/… for an example of the use of the technique described in Robert Israel’s answer.
$endgroup$
– amd
Apr 1 at 17:59
add a comment |
$begingroup$
Please bear with me.
Imagine the unit square in the plane to be a carrom board. Assume the striker is just
a point, moving with no friction (so it goes forever), and that when it hits an edge, the
angle of reflection is equal to the angle of incidence, as in real life. If the striker ever
hits a corner it falls into the pocket and disappears. The trajectory of the striker is
completely determined by its starting point (x, y) and its initial velocity $$vec(p, q)$$.
If the striker eventually returns to its initial state (i.e. initial position and initial velocity),
we define its bounce number to be the number of edges it hits before returning to its initial
state for the first time.
For example, the traectory with initial state $$[(.5, .5);vec(1, 0)]$$ has bounce number 2 and
it returns to its initial state for the first time in 2 time units. And the trajectory with
initial state $$[(.25, .75); vec(1, 1)] $$has bounce number 4.
(a) Suppose the striker has initial state $[(.5, .5);vec(p, q)]$. If p > q ≥ 0 then what is the
velocity after it hits an edge for the first time? What if q > p ≥ 0?
(b) Draw a trajectory with bounce number 5 or justify why it is impossible.
(c) Consider the trajectory with initial state $[(x, y); vec(p, 0)]$ where p is a positive integer.
In how much time will the striker first return to its initial state?
(d) What is the bounce number for the initial state $[(x, y);vec
(p, q)]$ where p, q are relatively
prime positive integers, assuming the striker never hits a corner?
I have done (a), (b) and (c) and am stuck on (d).
Answer to (a) is $vec(-p,q)$ and $vec(p,-q)$.
Answer to (c) is $$frac2p$$
Can you please tell me how to do it?
Thanks.
vectors
asked Apr 1 at 17:32
RaghavRaghav
627
$endgroup$
Please bear with me.
Imagine the unit square in the plane to be a carrom board. Assume the striker is just
a point, moving with no friction (so it goes forever), and that when it hits an edge, the
angle of reflection is equal to the angle of incidence, as in real life. If the striker ever
hits a corner it falls into the pocket and disappears. The trajectory of the striker is
completely determined by its starting point (x, y) and its initial velocity $$vec(p, q)$$.
If the striker eventually returns to its initial state (i.e. initial position and initial velocity),
we define its bounce number to be the number of edges it hits before returning to its initial
state for the first time.
For example, the traectory with initial state $$[(.5, .5);vec(1, 0)]$$ has bounce number 2 and
it returns to its initial state for the first time in 2 time units. And the trajectory with
initial state $$[(.25, .75); vec(1, 1)] $$has bounce number 4.
(a) Suppose the striker has initial state $[(.5, .5);vec(p, q)]$. If p > q ≥ 0 then what is the
velocity after it hits an edge for the first time? What if q > p ≥ 0?
(b) Draw a trajectory with bounce number 5 or justify why it is impossible.
(c) Consider the trajectory with initial state $[(x, y); vec(p, 0)]$ where p is a positive integer.
In how much time will the striker first return to its initial state?
(d) What is the bounce number for the initial state $[(x, y);vec
(p, q)]$ where p, q are relatively
prime positive integers, assuming the striker never hits a corner?
I have done (a), (b) and (c) and am stuck on (d).
Answer to (a) is $vec(-p,q)$ and $vec(p,-q)$.
Answer to (c) is $$frac2p$$
Can you please tell me how to do it?
Thanks.
vectors
vectors
asked Apr 1 at 17:32
RaghavRaghav
627
asked Apr 1 at 17:32
RaghavRaghav
627
asked Apr 1 at 17:32
RaghavRaghav
627
asked Apr 1 at 17:32
RaghavRaghav
627
asked Apr 1 at 17:32
RaghavRaghav
627
627
$begingroup$
See math.stackexchange.com/questions/2800897/… for an example of the use of the technique described in Robert Israel’s answer.
$endgroup$
– amd
Apr 1 at 17:59
add a comment |
$begingroup$
See math.stackexchange.com/questions/2800897/… for an example of the use of the technique described in Robert Israel’s answer.
$endgroup$
– amd
Apr 1 at 17:59
$begingroup$
See math.stackexchange.com/questions/2800897/… for an example of the use of the technique described in Robert Israel’s answer.
$endgroup$
– amd
Apr 1 at 17:59
$begingroup$
See math.stackexchange.com/questions/2800897/… for an example of the use of the technique described in Robert Israel’s answer.
$endgroup$
– amd
Apr 1 at 17:59
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The usual way to do this sort of problem is using images. For example, here we have a straight line from the centre of one lattice cell to another crossing $5$ edges.
This corresponds to a trajectory that goes from the centre of your square to itself after $5$ reflections.
answered Apr 1 at 17:56
Robert IsraelRobert Israel
332k23222481
$endgroup$
add a comment |
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$begingroup$
The usual way to do this sort of problem is using images. For example, here we have a straight line from the centre of one lattice cell to another crossing $5$ edges.
This corresponds to a trajectory that goes from the centre of your square to itself after $5$ reflections.
answered Apr 1 at 17:56
Robert IsraelRobert Israel
332k23222481
$endgroup$
add a comment |
$begingroup$
The usual way to do this sort of problem is using images. For example, here we have a straight line from the centre of one lattice cell to another crossing $5$ edges.
This corresponds to a trajectory that goes from the centre of your square to itself after $5$ reflections.
answered Apr 1 at 17:56
Robert IsraelRobert Israel
332k23222481
$endgroup$
add a comment |
$begingroup$
The usual way to do this sort of problem is using images. For example, here we have a straight line from the centre of one lattice cell to another crossing $5$ edges.
This corresponds to a trajectory that goes from the centre of your square to itself after $5$ reflections.
answered Apr 1 at 17:56
Robert IsraelRobert Israel
332k23222481
$endgroup$
The usual way to do this sort of problem is using images. For example, here we have a straight line from the centre of one lattice cell to another crossing $5$ edges.
This corresponds to a trajectory that goes from the centre of your square to itself after $5$ reflections.
answered Apr 1 at 17:56
Robert IsraelRobert Israel
332k23222481
answered Apr 1 at 17:56
Robert IsraelRobert Israel
332k23222481
answered Apr 1 at 17:56
Robert IsraelRobert Israel
332k23222481
answered Apr 1 at 17:56
Robert IsraelRobert Israel
332k23222481
332k23222481
add a comment |
add a comment |
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$begingroup$
See math.stackexchange.com/questions/2800897/… for an example of the use of the technique described in Robert Israel’s answer.
$endgroup$
– amd
Apr 1 at 17:59