Trajectory
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 cannonball, 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 second law,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \frac{d^2\vec{x}}{dt^2} = \vec{F} }
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F}} 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 cannonball. 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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_x = v \cos{\theta}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_y = u \sin{\theta}} . The x and y positions of the particle can be expressed as:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t) = u_x t = u \cos{\theta} t }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t) = u_y t - \frac{1}{2}gt^2 = u \sin{\theta} t - \frac{1}{2}gt^2 }
These can be rearranged so that the trajectory followed is:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x) = x \tan{\theta} - \frac{g}{2u^2 \cos^2{\theta}}x^2 }
Hence the path followed by a body such as cannonball is roughly 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:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{max} = \frac{2u^2 \cos^2{\theta} \tan{\theta}}{g} = \frac{u^2 \sin{2\theta}}{g} }
The maximum height can be found as:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{max} = \frac{u^2 \sin^2{\theta}}{2g} }