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The first solution corresponds to when the projectile is first launched. The second solution is the useful one for determining the range of the projectile. Plugging this value for (t) into the horizontal equation yields = Applying the trigonometric identity
A solution of the falling cat problem is a curve in the configuration space that is horizontal with respect to the connection (that is, it is admissible by the physics) with prescribed initial and final configurations. Finding an optimal solution is an example of optimal motion planning. [11] [12]
In projectile motion, the horizontal motion and the vertical motion are independent of each other; that is, neither motion affects the other. This is the principle of compound motion established by Galileo in 1638, [ 1 ] and used by him to prove the parabolic form of projectile motion.
The curve of fastest descent is not a straight or polygonal line (blue) but a cycloid (red).. In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'), [1] or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides ...
Later they extended the technique to many other cases, for example, to 3D open-chain kinematic robots under full Lagrangian dynamics. [2] [3] More recently, many practical heuristic algorithms based on stochastic optimization and iterative sampling were developed, by a wide range of authors, to address the kinodynamic planning problem. These ...
A commonly used solution to the problem exists for n = 3 called P3P, and many solutions are available for the general case of n ≥ 3. A solution for n = 2 exists if feature orientations are available at the two points. [3] Implementations of these solutions are also available in open source software.
Finding the shortest solution sequence in the pebble motion on graphs problem (with labeled pebbles) is known to be NP-hard [6] and APX-hard. [3] The unlabeled problem can be solved in polynomial time when using the cost metric mentioned above (minimizing the total number of moves to adjacent vertices), but is NP-hard for other natural cost ...
It is difficult (but not completely impossible) to separate the physical simulation from the collision detection algorithm. However, in all but the simplest cases, the problem of determining ahead of time when two bodies will collide (given some initial data) has no closed form solution—a numerical root finder is usually involved.