Equation of Trajectory
The trajectory of an object in motion can be described by an equation that relates the object's position (x,y) at any given time t. This equation is typically derived based on the forces acting on the object and the initial conditions of its motion.

Key Points:
- The equation of trajectory is often used in physics to analyze the motion of projectiles, such as a ball thrown through the air.
- It is a mathematical representation of the path that an object follows as it moves through space.
- The trajectory equation can be expressed in parametric form, with separate equations for x(t) and y(t) representing the horizontal and vertical positions of the object as functions of time.

Derivation:
The trajectory equation is derived by considering the forces acting on the object, such as gravity, air resistance, and initial velocity. By using Newton's laws of motion and kinematic equations, the equations of motion can be determined. These equations are then solved to obtain the trajectory equation in terms of x(t) and y(t).

Example:
For a projectile launched at an angle θ with an initial velocity v, the trajectory equation can be expressed as:
x(t) = v*cos(θ)*t
y(t) = v*sin(θ)*t - (1/2)*g*t^2
where g is the acceleration due to gravity. This set of equations describes the horizontal and vertical positions of the projectile as functions of time, allowing us to analyze its motion and predict its trajectory.
In conclusion, the equation of trajectory is a fundamental concept in physics that helps us understand the motion of objects in space. It provides a mathematical framework for analyzing and predicting the path of projectiles and other moving objects.