In kinematics, we study the motion of various objects and the path taken by the object while changing its position by the effect of a force applied to it. the path can be regular and quantifiable like a straight line or can be irregular without any definite direction. The other parameters such as speed and acceleration are also measured by observing the different values of the object while in motion. We also have studied the attraction force that the earth exerts on every object over its surface. It is known as the force of gravity and is represented by the symbol ‘g’. Thus, any object moving near the surface of the earth is influenced by this force and gets accelerated downward towards the ground. the object in such a condition is said to be in a projectile motion.
The path that it attains is known as the trajectory. The best example of a projectile motion is when you throw a stone into the air. It travels to a certain height by the force provided by you and then falls freely under the influence of gravity. But the horizontal flight of the object remains unchanged and we can observe a curved path of motion. For the same speed, it travels the maximum distance when we throw a stone at 45 degrees angle to the ground. A batsman in the game of cricket also tries to hit the ball 45 degrees to get the ball away from the boundary. The same principle is used in various other areas of application. Ballistic missiles that we hear about all the time in the news also use this principle to travel the maximum distance possible during wars.
The equations to find out the maximum distance, maximum height and time of flight are mentioned in the chapters of motion.
Trajectory Equation
A trajectory or flight path in physics is defined as the path that an object in motion having some mass follows through space as a function of time. Hamiltonian mechanics via canonical coordinates are used to define the trajectory of an object in classical mechanics. Hence both position and momentum simultaneously define a complete trajectory. For example, the motion of a ball or a big rock was thrown upwards or the motion of a satellite or a bullet fired from a gun.
Derivation of Equation of Trajectory
The equation of trajectory derivation is as follows —
[ y = x~tan~theta – frac{gx^{2}}{2 v^{2} cos^{2} theta}]
where, ‘x’ is the horizontal component of the trajectory.
‘y’ is the vertical component of the trajectory.
‘g’ is the gravitational force of attraction.
‘v’ is the initial velocity of the object.
‘ϴ’ is the angle of elevation of the trajectory.
Some Important Trajectory Related Equations are as Follows
Time of Flight,[T = frac {2v_{0} sintheta} {g}]
Maximum Height Reached, [H = frac{v{_{0}}^{2} sin^{2} Theta}{2g}]
Horizontal Range , [R = frac{v{_{0}}^{2} sin2 Theta}{g}]
Where ‘[v_{0}]’ is the initial velocity
‘sinϴ’ is the vertical component of the y-axis
‘cosϴ’ is the horizontal component of the x-axis.
Thus the trajectory equation along with some important formulae has been derived. Objects will follow a vertically symmetric path when projected from and land on the same horizontal surface. The flight time depends on the initial velocity of the projectile and the angle of projection. The maximum height of the projectile is reached when the velocity of the object is zero. The range of the projectile depends on the initial velocity of the object.
This article explains the trajectory formula and the derivation of the equation of trajectory. Some FAQs have been added for a better understanding of the topic for the students.