Vector analysis: show the object moves on the elliptical path

In summary: Since x and y are determined by a single parameter, t, the object moves along a curve in the plane. As you say, the curve is given by the equation ##(\frac{x}{a})^2+ (\frac{y}{b})^2= 1##.In summary, the given problem is to show that an object moves on an elliptical path in the xy-plane, with its position given by the equation r= i acosωt+j bsinωt. The relevant equations are (x/a)^2+(y/b)^2=1 and the attempt at a
  • #1
manimaran1605
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0

Homework Statement


problem form div curl grad and all that by h.m.schey
An object moves in xy-plane in such a way that its position vector r is given as a some function of "t" by

r= i acosωt+j bsinωt

show the object moves on the elliptical path

Homework Equations


(x/a)^2+(y/b)^2=1

The Attempt at a Solution


I tried the problem by changing cosωt=x and sinωt=y but i don't know how to proceed further, i don't know how to eliminate the unit vectors i, j. help me. Thank you
 
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  • #2
You have to show that for the given parametrisation of the coordinates ##x(t)## and ##y(t)##, the equation for an ellipse is satisfied, that equation being the one you wrote in Relevant equations. What is x(t) and y(t) in this case?
 
  • #3
manimaran1605 said:

Homework Statement


problem form div curl grad and all that by h.m.schey
An object moves in xy-plane in such a way that its position vector r is given as a some function of "t" by

r= i acosωt+j bsinωt

show the object moves on the elliptical path

Homework Equations


(x/a)^2+(y/b)^2=1


The Attempt at a Solution


I tried the problem by changing cosωt=x and sinωt=y
Well, that's your first error. It should be x= a cos(ωt), y= b sin(ωt) so that x/a= cos(ωt), y/b= sin(ωt). Now use the fact that sin^2+ cos^2= 1.

but i don't know how to proceed further, i don't know how to eliminate the unit vectors i, j. help me. Thank you
?? There are no vectors in your equation for x and y as functions of t!
 

1. What is vector analysis?

Vector analysis is a mathematical method used to analyze and describe the movement of objects in a three-dimensional space. It involves the use of vectors, which are quantities that have both magnitude and direction.

2. How is vector analysis used to show an object's movement on an elliptical path?

In vector analysis, the position, velocity, and acceleration of an object are represented by vectors. By analyzing these vectors, we can determine the direction and magnitude of an object's movement on an elliptical path.

3. What is an elliptical path?

An elliptical path is a type of curved path that resembles an ellipse. It is a closed curve with two distinct points called foci, where the sum of the distances from any point on the curve to the two foci remains constant. It is a common path for objects moving under the influence of a central force, such as gravity.

4. What factors affect an object's movement on an elliptical path?

The factors that affect an object's movement on an elliptical path include the object's mass, the magnitude and direction of the central force acting on it, and the initial velocity and position of the object.

5. How is vector analysis used in real-world applications?

Vector analysis has many real-world applications, including physics, engineering, and computer graphics. It is used to describe and analyze the motion of objects in various fields, such as projectile motion, planetary motion, and fluid flow. It is also used in the design of structures and machines, as well as in the creation of computer-generated images and animations.

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