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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
In this equation, the origin is the midpoint of the horizontal range of the projectile, and if the ground is flat, the parabolic arc is plotted in the range . This expression can be obtained by transforming the Cartesian equation as stated above by y = r sin ϕ {\displaystyle y=r\sin \phi } and x = r cos ϕ {\displaystyle x=r\cos \phi } .
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.
Horizontal and vertical lines. In the general equation of a line, ax + by + c = 0, a and b cannot both be zero unless c is also zero, in which case the equation does not define a line. If a = 0 and b ≠ 0, the line is horizontal and has equation y = -c/b.
The general equation's coefficients can be obtained from known semi-major axis , semi-minor axis , center coordinates (,), and rotation angle (the angle from the positive horizontal axis to the hyperbola's major axis) using the formulae:
The direction of the velocity vector when projected on to the horizontal will be called the track, denoted . The body orientation is called the heading, denoted (psi). The force equation of motion includes a component of weight: [citation needed]
If the horizontal cross-section moves up or down, toward or away from the apex of the cone, D and E move along the parabola, always maintaining the relationship between x and y shown in the equation. The parabolic curve is therefore the locus of points where the equation is satisfied, which makes it a Cartesian graph of the quadratic function ...
Torricelli's law is obtained as a special case when the opening is very small relative to the horizontal cross-section of the container : v A = 2 g h . {\displaystyle v_{A}={\sqrt {2gh}}.} [ 1 ] Torricelli's law can only be applied when viscous effects can be neglected which is the case for water flowing out through orifices in vessels.