Mixing
“Slippery Slope” - a Parametric Trajectory problem
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I'm designing some equipment that spits out a small ball, which I want to capture when it lands, without it bouncing away. I figure this can be done by having the ball strike a slope tangentially. I define the problem as:
"A particle P subject to earth's gravity leaves a launch point (0,k) at positive velocity V in both the x and y direction. Is there a slope S, lying below and to the right of k, such that for any V, the particle hits S tangentially ? If so, express Sy as a function of Sx"
Diagram shows x and y axes (black), Particle P's initial velocity vectors (red), and subsequent parabolic motion (blue, depending on V). Curve S (green) is formed when P glances tangentially regardless of V.
Forming the parametric equations for Px and Py as functions of t (time) is simple. Px and Py can then be expressed as functions of V. Conceptually, the constraint that the slope of S equals the slope of P at the point of impact should be incorporated as a constraint, but not sure how to do this.
Appreciate all assistance
calculus vector-analysis mathematical-physics parametric
asked Jan 30 at 9:32
DanDan
11
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add a comment |
$begingroup$
I'm designing some equipment that spits out a small ball, which I want to capture when it lands, without it bouncing away. I figure this can be done by having the ball strike a slope tangentially. I define the problem as:
"A particle P subject to earth's gravity leaves a launch point (0,k) at positive velocity V in both the x and y direction. Is there a slope S, lying below and to the right of k, such that for any V, the particle hits S tangentially ? If so, express Sy as a function of Sx"
Diagram shows x and y axes (black), Particle P's initial velocity vectors (red), and subsequent parabolic motion (blue, depending on V). Curve S (green) is formed when P glances tangentially regardless of V.
Forming the parametric equations for Px and Py as functions of t (time) is simple. Px and Py can then be expressed as functions of V. Conceptually, the constraint that the slope of S equals the slope of P at the point of impact should be incorporated as a constraint, but not sure how to do this.
Appreciate all assistance
calculus vector-analysis mathematical-physics parametric
asked Jan 30 at 9:32
DanDan
11
$endgroup$
$begingroup$
Does "any $V$" mean the vectors that have the same length as $V$ but different directions? Because I think adjusting the length (speed) of the initial vector makes it impossible to have a common curve S ...
$endgroup$
– Matti P.
Jan 30 at 10:13
$begingroup$
Another question: Why can we assume that there is a commons curve (slope) S for the different trajectories? The range of a thrown ball, for example (en.wikipedia.org/wiki/Trajectory#Range_and_height), does depend on the initial angle ...
$endgroup$
– Matti P.
Jan 30 at 10:15
$begingroup$
All initial velocity vectors V have the same initial direction (45 degrees), just different magnitudes. The trajectory of P for each V will differ and have a different slope at the impact point with S. The question asks whether S, which is the curve formed by all such impact points, exists.
$endgroup$
– Dan
Jan 31 at 11:43
add a comment |
$begingroup$
I'm designing some equipment that spits out a small ball, which I want to capture when it lands, without it bouncing away. I figure this can be done by having the ball strike a slope tangentially. I define the problem as:
"A particle P subject to earth's gravity leaves a launch point (0,k) at positive velocity V in both the x and y direction. Is there a slope S, lying below and to the right of k, such that for any V, the particle hits S tangentially ? If so, express Sy as a function of Sx"
Diagram shows x and y axes (black), Particle P's initial velocity vectors (red), and subsequent parabolic motion (blue, depending on V). Curve S (green) is formed when P glances tangentially regardless of V.
Forming the parametric equations for Px and Py as functions of t (time) is simple. Px and Py can then be expressed as functions of V. Conceptually, the constraint that the slope of S equals the slope of P at the point of impact should be incorporated as a constraint, but not sure how to do this.
Appreciate all assistance
calculus vector-analysis mathematical-physics parametric
asked Jan 30 at 9:32
DanDan
11
$endgroup$
I'm designing some equipment that spits out a small ball, which I want to capture when it lands, without it bouncing away. I figure this can be done by having the ball strike a slope tangentially. I define the problem as:
"A particle P subject to earth's gravity leaves a launch point (0,k) at positive velocity V in both the x and y direction. Is there a slope S, lying below and to the right of k, such that for any V, the particle hits S tangentially ? If so, express Sy as a function of Sx"
Diagram shows x and y axes (black), Particle P's initial velocity vectors (red), and subsequent parabolic motion (blue, depending on V). Curve S (green) is formed when P glances tangentially regardless of V.
Forming the parametric equations for Px and Py as functions of t (time) is simple. Px and Py can then be expressed as functions of V. Conceptually, the constraint that the slope of S equals the slope of P at the point of impact should be incorporated as a constraint, but not sure how to do this.
Appreciate all assistance
calculus vector-analysis mathematical-physics parametric
calculus vector-analysis mathematical-physics parametric
asked Jan 30 at 9:32
DanDan
11
asked Jan 30 at 9:32
DanDan
11
asked Jan 30 at 9:32
DanDan
11
asked Jan 30 at 9:32
DanDan
11
asked Jan 30 at 9:32
DanDan
11
11
$begingroup$
Does "any $V$" mean the vectors that have the same length as $V$ but different directions? Because I think adjusting the length (speed) of the initial vector makes it impossible to have a common curve S ...
$endgroup$
– Matti P.
Jan 30 at 10:13
$begingroup$
Another question: Why can we assume that there is a commons curve (slope) S for the different trajectories? The range of a thrown ball, for example (en.wikipedia.org/wiki/Trajectory#Range_and_height), does depend on the initial angle ...
$endgroup$
– Matti P.
Jan 30 at 10:15
$begingroup$
All initial velocity vectors V have the same initial direction (45 degrees), just different magnitudes. The trajectory of P for each V will differ and have a different slope at the impact point with S. The question asks whether S, which is the curve formed by all such impact points, exists.
$endgroup$
– Dan
Jan 31 at 11:43
add a comment |
$begingroup$
Does "any $V$" mean the vectors that have the same length as $V$ but different directions? Because I think adjusting the length (speed) of the initial vector makes it impossible to have a common curve S ...
$endgroup$
– Matti P.
Jan 30 at 10:13
$begingroup$
Another question: Why can we assume that there is a commons curve (slope) S for the different trajectories? The range of a thrown ball, for example (en.wikipedia.org/wiki/Trajectory#Range_and_height), does depend on the initial angle ...
$endgroup$
– Matti P.
Jan 30 at 10:15
$begingroup$
All initial velocity vectors V have the same initial direction (45 degrees), just different magnitudes. The trajectory of P for each V will differ and have a different slope at the impact point with S. The question asks whether S, which is the curve formed by all such impact points, exists.
$endgroup$
– Dan
Jan 31 at 11:43
$begingroup$
Does "any $V$" mean the vectors that have the same length as $V$ but different directions? Because I think adjusting the length (speed) of the initial vector makes it impossible to have a common curve S ...
$endgroup$
– Matti P.
Jan 30 at 10:13
$begingroup$
Does "any $V$" mean the vectors that have the same length as $V$ but different directions? Because I think adjusting the length (speed) of the initial vector makes it impossible to have a common curve S ...
$endgroup$
– Matti P.
Jan 30 at 10:13
$begingroup$
Another question: Why can we assume that there is a commons curve (slope) S for the different trajectories? The range of a thrown ball, for example (en.wikipedia.org/wiki/Trajectory#Range_and_height), does depend on the initial angle ...
$endgroup$
– Matti P.
Jan 30 at 10:15
$begingroup$
Another question: Why can we assume that there is a commons curve (slope) S for the different trajectories? The range of a thrown ball, for example (en.wikipedia.org/wiki/Trajectory#Range_and_height), does depend on the initial angle ...
$endgroup$
– Matti P.
Jan 30 at 10:15
$begingroup$
All initial velocity vectors V have the same initial direction (45 degrees), just different magnitudes. The trajectory of P for each V will differ and have a different slope at the impact point with S. The question asks whether S, which is the curve formed by all such impact points, exists.
$endgroup$
– Dan
Jan 31 at 11:43
$begingroup$
All initial velocity vectors V have the same initial direction (45 degrees), just different magnitudes. The trajectory of P for each V will differ and have a different slope at the impact point with S. The question asks whether S, which is the curve formed by all such impact points, exists.
$endgroup$
– Dan
Jan 31 at 11:43
add a comment |
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$begingroup$
Does "any $V$" mean the vectors that have the same length as $V$ but different directions? Because I think adjusting the length (speed) of the initial vector makes it impossible to have a common curve S ...
$endgroup$
– Matti P.
Jan 30 at 10:13
$begingroup$
Another question: Why can we assume that there is a commons curve (slope) S for the different trajectories? The range of a thrown ball, for example (en.wikipedia.org/wiki/Trajectory#Range_and_height), does depend on the initial angle ...
$endgroup$
– Matti P.
Jan 30 at 10:15
$begingroup$
All initial velocity vectors V have the same initial direction (45 degrees), just different magnitudes. The trajectory of P for each V will differ and have a different slope at the impact point with S. The question asks whether S, which is the curve formed by all such impact points, exists.
$endgroup$
– Dan
Jan 31 at 11:43