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Line 1: − As part of the Betha [[Chemistry]] Tutorial created by the [[Ohio State University]]'s Department of Chemistry, the following explanation was given: + − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + − ==References== − <references/> +
Added information about projectile motion in uniform gravitational field, max range and height
The '''trajectory''' of an object is the path it takes through space. It is often described by the [[position]] of an object as a function of time. An example is that of a cannon ball, but it applies to any path such as the [[orbit]] of a [[planet]] or a rocket in space. In [[classical mechanics]], the trajectory of a particle with mass m is described by [[Newton's Laws of Motion|Newton's second law]],
{{quotebox|The x, y and z coordinates of a particle as a function of time are known as the trajectory or orbit of a particle. The laws of classical physics predict the trajectory of a particle for all times once the position and velocity are known at some initial time. For example, if the position and velocity of a cannonball are known at the instant it leaves a cannon, the classical mechanics can predict the path taken by the cannonball at later times and where it will land.<ref>[http://www.chemistry.ohio-state.edu/betha/qm/1bfrb.html "An Introduction to Quantum Mechanics" at Ohio State University]</ref>}}
<math>
m \frac{d^2\vec{x}}{dt^2} = \vec{F}
</math>
where <math>\vec{F}</math> is the net [[force]] that acts on the particle.
==Projectile motion==
A useful example of trajectories is that of projectile motion, such as the motion of a cannon ball. The simplest case is that of 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 the particle in the x and y directions are independent, the motion in each dimension can be considered separately. For a body with initial speed, u, fired at an angle θ above the horizontal, the x an y components of the body's [[velocity]] can 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 followed by a body such as cannon ball is 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:
<math>
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://hyperphysics.phy-astr.gsu.edu/hbase/traj.html Trajectory at Hyperphysics]
[[Category:Physics]]
[[Category:Physics]]
[[Category:Mechanics]]