Equation of Trajectory
A trajectory is described as a position of an object over a particular time. It is the path taken up by a moving object that is following through the space as a function of time. A bullet fired from gun is an example of trajectory motion. A projectile is thrown with velocity u at an angle θ with the horizontal. The velocity u can be resolved into two rectangular components: u cosθ component along X – axis and u sinθ component along Y – axis.
For horizontal motion:
x = u cosθ x t
\(t=\frac{x}{u\cos \theta }\) … (1)
For vertical motion:
y = (u sinθ) t – ½ gt² … (2)
From equations (1) and (2):
\(y=u\sin \theta \left( \frac{x}{u\cos \theta } \right)-\frac{1}{2}g\left( \frac{{{x}^{2}}}{{{u}^{2}}{{\cos }^{2}}\theta } \right)\).
\(y=x\tan \theta -\frac{1}{2}\frac{g{{x}^{2}}}{{{u}^{2}}{{\cos }^{2}}\theta }\) … (3)
This equation shows that the trajectory of projectile is parabolic because it is similar to the equation of parabola.
y = ax – bx²
It is known as the equation of trajectory. It is an equation of parabola. Hence, path of a projector is parabolic. Equation (3) can also be written as:
\(y=x\tan \theta -\frac{{{x}^{2}}}{\frac{2{{u}^{2}}{{\cos }^{2}}\theta }{g}}\).
∴ Equation of Trajectory (y) =\(x\tan \theta -\frac{g{{x}^{2}}}{\left(
2{{u}^{2}}{{\cos }^{2}}\theta\right)}\).
Where,
y = Vertical Component,
x = Horizontal Component,
g = Acceleration due to gravity,
u = Initial Velocity,
θ = Angle of inclination of the initial velocity from horizontal axis.