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This is the equation of a parabola, so the path is parabolic. The axis of the parabola is vertical. If the projectile's position (x,y) and launch angle (θ or α) are known, the initial velocity can be found solving for v 0 in the afore-mentioned parabolic equation:
For a parametric equation of a parabola in general position see § As the affine image of the unit parabola. The implicit equation of a parabola is defined by an irreducible polynomial of degree two: + + + + + =, such that =, or, equivalently, such that + + is the square of a linear polynomial.
The green path in this image is an example of a parabolic trajectory. A parabolic trajectory is depicted in the bottom-left quadrant of this diagram, where the gravitational potential well of the central mass shows potential energy, and the kinetic energy of the parabolic trajectory is shown in red. The height of the kinetic energy decreases ...
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
A three-dimensional version of parabolic coordinates is obtained by rotating the two-dimensional system about the symmetry axis of the parabolas. Parabolic coordinates have found many applications, e.g., the treatment of the Stark effect and the potential theory of the edges.
A complete Fermat's spiral (both branches) is a smooth double point free curve, in contrast with the Archimedean and hyperbolic spiral. Like a line or circle or parabola, it divides the plane into two connected regions. Definition of sector (light blue) and polar slope angle α
The paraboloid of revolution obtained by rotating the safety parabola around the vertical axis is the boundary of the safety zone, consisting of all points that cannot be hit by a projectile shot from the given point with the given speed.
We consider the following example of envelope in motion. Suppose at initial height 0, one casts a projectile into the air with constant initial velocity v but different elevation angles θ. Let x be the horizontal axis in the motion surface, and let y denote the vertical axis. Then the motion gives the following differential dynamical system:
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