Not a blog
Natural and Imaginary world is full of amazing phenomena. For example, water shooting up periodically from a geyser in Iceland. Using concepts of kinematics we can estimate the maximum velocity of water particles at the mouth of the geyser by measuring the height to which the water reaches above ground.
Prediction of trajectory of a Space X Rocket succesfully returning to Earth requires application of the ideas which we are going to study next.
Prerequisites for the lesson: If you have beginner level understanding of vectors you can go ahead, otherwise you can watch my lecture on vectors. Please watch the lecture on 1.5X speed as I construct ideas rather slowly.
Frame of Refrence
A frame of refrence is a set of three mutually perpendicular axes which intersect at origin (named O), alongwith a clock to measure time. For Competitive examinations or Boards, we commonly use the same coordinate system which you might have studied in Class X. The frame of refrence can be attatched to a particle or to a rigid body or fixed in space and therefore provide an “absolute” point of view when studying motion of other objects.
Position
The coordinates of the particle (x,y,z) relative to the origin (O) is the position of the particle. The values x, y and z represent the distance to be moved from the origin along the respective axis to reach the particle. For example, (2,3,9)m represents the particle can be found by moving 2m along x axis, 3m along y axis, and 9m along z axis, in no particular order (i.e. any order). Position of a particle can be represented as a vector, where if (x,y,z) is the position of the particle then the corresponding vector is,
For example the position vector or (simply) the position of an imaginary star at (2,3,9)ly (ly=light year) as seen by the Sun is,
The Position of a particle moving on a straight line can be represented by x, where x > 0 represents the particle existing on the positive side and x < 0 represents the particle existing on the negative side of the origin ( x = 0 ).
The position of the particle can be a “function” of time, which is a fancy (or mathematical) way of saying that the position of the particle depends on / varies with time.
In the above equation, if we use t=1 then we will get the position of the particle at that time.
we can substitute any other value of t to get the the corresponding position of the particle. Try to find the time at which the particle whose position is given below crosses origin.
Using guesswork, we can say that the particle would cross the origin at t=1. Perhaps their is some other way to do this excercise without just guessing.
Next, try to find the time at which the particle whose position is given below crosses origin.
At first we observe that the unit vectors in the expression are not grouped. So the first step should be to group them to reach an intermediate stage.
Now each expression within the bracket can be equated with the corresponding coordinate to get the time(s) at which the value of expression would be equal to the coordinate.
We can see that even though the motion of the particle is not easy to imagine we can find the time at which the particle can reach a certain position, if at all.
Distance
The total path length of the path traversed by a particle in going from initial to final position is called the Distance covered by the particle.
Find the distance covered by the particle in going from P to Q along the circle. And, find the distance covered by the particle by the particle in going from P to Q along the diameter.
Displacement
Displacement of a particle is defined as the shortest distance between initial and final position, alongwith a direction from the initial to final position. Displacement of a particle in going from initial to final position is independent of the path taken, which is beacuse of the way it is defined. In the example drawn below a particle goes from r1 to r2, and the displacement is equal to r2 - r1.
For example, if the position of a particle is x = 2t unit, where t is time (in second) then the displacement of the particle between 2s and 4s is x(final) - x(initial) = 2 * 4 - 2 * 2 = 4 unit. Akin to position, displacement of a particle moving in straight line can be either positive or negetive. Generalization of displacement in two dimensions,
Take an example,
The displacement of a particle is made up of two components, the magnitude and the direction. The direction of the displacement could simply be something like North, 30° East of South etc., or it can be represented as follows,
A twist in subjective IIT-JEE Question
Find the ratio of the magnitude of the displacement and the distance covered by the particle as it goes along the path ABC from A to C.
