what is trajectory

what is trajectory

A trajectory is the path followed by an object with mass in curvilinear motion (curved path)  as a function of time.

Mathematically ,It is defined by the equation in x-y coordinates.

A famous example would be projectile motion like the  ball thrown an angle  or a bullet fired from gun.  Here the path traveled by the ball or bullet followed an curvillinear motion under the influence of gravitational force. The path of the motion of the ball or bullet is the trajectory of the ball or bullet.  Here we know that  trajectory in these cases is a  parabola

How to Find the trajectory of the motion

Example -1

A object moves such that its x and y coordinates varies as given below

$x=at$  -(1)

$y=b -ct$  -(2)

Here the trajectory of the object can be found by eliminating time from the equation

From equation (1)

$t= \frac {x}{a}$

Substituting these values in equation (2), we get

$y=b – c(\frac {x}{a})$

The above equation is called the trajectory of the object

Example-2 

Motion of the projectile motion

X-coordinate at any point

$x=(v_0cos{\theta _0})t$  -(1)

Y-coordinate at any point
$y=(v_0sin{\theta _0})t- \frac {1}{2}gt^2$  –(2)



Here the trajectory of the object can be found by eliminating time from the equation

From equation (1)

$t=\frac {x}{(v_0cos{\theta _0})}$

Substituting these values in equation (2), we get

$y=(tan{\theta _0})x-[\frac {g}{2(v_0cos{\theta _0})^2}]x^2$

Example -3

A object moves such that its position vector varies as

$R=(Asin \omega t)i+(Acos \omega t)j$

So $x=Acos \omega t$  -(1)

$cos \omega t=\frac {x}{A}$

$y=Asin \omega t$  -(2)
$sin \omega t=\frac {y}{A}$

Now we know that

$Sin^2 \omega t+cos^2 \omega t=1$

So, $(\frac {x}{A})^2+(\frac {y}{A})^2=1$
$x^2+y^2=A^2$

Which is a equation of a circle

In Physics it is important to know what is motion? Motion and its study lay the basic foundation for the student who wants to study physics. It is important for students to master these concepts.
If you want to dive further and learn more about motion and kinematics you can consider buying the book
Interactive Physics Part 3.
This book covers most portions of Mechanics. It gives you a thorough grounding in Physics and imparts skills so that you can tackle IIT JEE, Engineering and Medical entrance exams with confidence. The application methods and concepts/misconceptions can be applied to various entrance examinations (IIT JEE, JEE Main, NEET, AIIMS, JIPMER, BITSAT etc.)
For those of you who are interested in video tutorials can look at this video course by udemy
Mastering Kinematics
This course will guide you to solve any, one and two-dimensional kinematics problem, using kinematics equations and solving graphical problems. Through this course, you can learn how to solve relative motion problems such as river crossing problems


DISCLOSURE: THIS POST MAY CONTAIN AFFILIATE LINKS, MEANING I GET A COMMISSION IF YOU DECIDE TO MAKE A PURCHASE THROUGH MY LINKS, AT NO COST TO YOU. PLEASE READ MY DISCLOSURE FOR MORE INFO.



Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

  Subscribe  
Notify of