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As part The '''trajectory''' of an object is the path it takes through space. It is often described by the Betha [[Chemistryposition]] Tutorial created by of an object as a function of time. An example is that of a cannonball, but it applies to any path such as the [[Ohio State Universityorbit]]of a [[planet]] or a rocket in space. In [[classical mechanics]], the trajectory of a particle with mass m is described by [[Newton's Department Laws of ChemistryMotion|Newton's second law]], the following explanation was given:  <math>m \frac{d^2\vec{quotebox|The x, y and z coordinates of a }}{dt^2} = \vec{F}</math> where <math>\vec{F}</math> is the net [[force]] that acts on the particle as a function . ==Projectile motion== A useful example of time are known trajectories is that of projectile motion, such as the trajectory or orbit motion of a particlecannonball. The laws simplest case is that of classical physics predict where drag is ignored and the force of [[gravity]] on the projectile is taken to be constant. In this case, an exact solution for the trajectory may be found using the [[SUVAT equations]]. As the [[acceleration]] of a the particle for all times once in the position x and velocity y directions are known at some initial timeindependent, the motion in each dimension can be considered separately. For examplea body with initial speed, if the position and velocity of a cannonball are known u, fired at an angle θ above the instant it leaves a cannonhorizontal, the classical mechanics x an y components of the body's [[velocity]] can predict be split into components: <math>u_x = v \cos{\theta}</math> and <math>u_y = u \sin{\theta}</math>. The x and y positions of the particle can be expressed as: <math>x(t) = u_x t = u \cos{\theta} t</math> <math>y(t) = u_y t - \frac{1}{2}gt^2 = u \sin{\theta} t - \frac{1}{2}gt^2</math> These can be rearranged so that the trajectory followed is: <math>y(x) = x \tan{\theta} - \frac{g}{2u^2 \cos^2{\theta}}x^2</math> Hence the path taken followed by the a body such as cannonball at later times and where it will landis roughly [[quadratic equation|parabolic]].  ===Range and Maximum Height=== For the case of uniform gravity and no air resistance, the range of a body can be found by solving y = 0: <refmath>x_{max} = \frac{2u^2 \cos^2{\theta} \tan{\theta}}{g} = \frac{u^2 \sin{2\theta}}{g}</math> The maximum height can be found as: <math>y_{max} = \frac{u^2 \sin^2{\theta}}{2g}</math> ==See also==* [[Classical mechanics]]* [[SUVAT equations]] ==External links==* [http://www.chemistryhyperphysics.ohiophy-stateastr.gsu.edu/bethahbase/qm/1bfrbtraj.html "An Introduction to Quantum Mechanics" Trajectory at Ohio State UniversityHyperphysics]</ref>}}