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To find the angle giving the maximum height for a given speed calculate the derivative of the maximum height = / with respect to , that is = / which is zero when = / =. So the maximum height H m a x = v 2 2 g {\displaystyle H_{\mathrm {max} }={v^{2} \over 2g}} is obtained when the projectile is fired straight up.
The range and the maximum height of the projectile do not depend upon its mass. Hence range and maximum height are equal for all bodies that are thrown with the same velocity and direction. The horizontal range d of the projectile is the horizontal distance it has traveled when it returns to its initial height (=).
Successive parabolic interpolation is a technique for finding the extremum (minimum or maximum) of a continuous unimodal function by successively fitting parabolas (polynomials of degree two) to a function of one variable at three unique points or, in general, a function of n variables at 1+n(n+3)/2 points, and at each iteration replacing the "oldest" point with the extremum of the fitted ...
Maximum height can be calculated by absolute value of in standard form of parabola. It is given as H = | c | = u 2 2 g {\displaystyle H=|c|={\frac {u^{2}}{2g}}} Range ( R {\displaystyle R} ) of the projectile can be calculated by the value of latus rectum of the parabola given shooting to the same level.
Conversely, if it is required to find the point B for a particular area SAB, find point J from HJ and point B as before. By Book 1, Proposition 16, Corollary 6 of Newton's Principia, the speed of a body moving along a parabola with a force directed towards the focus is inversely proportional to the square root of the radius.
An elliptic paraboloid is shaped like an oval cup and has a maximum or minimum point when its axis is vertical. In a suitable coordinate system with three axes x , y , and z , it can be represented by the equation [ 1 ] z = x 2 a 2 + y 2 b 2 . {\displaystyle z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}.} where a and b are constants that ...
The endpoints (,) of the minor axis lie at the height of the asymptotes over/under the hyperbola's vertices. Either half of the minor axis is called the semi-minor axis, of length b . Denoting the semi-major axis length (distance from the center to a vertex) as a , the semi-minor and semi-major axes' lengths appear in the equation of the ...
Consider, for example, the one-parameter family of tangent lines to the parabola y = x 2. These are given by the generating family F(t,(x,y)) = t 2 – 2tx + y. The zero level set F(t 0,(x,y)) = 0 gives the equation of the tangent line to the parabola at the point (t 0,t 0 2).