What can you say about the motion of an object with velocity vector perpendicular to position vector

Ormezzani6cuu 2022-04-18 Answered
What can you say about the motion of an object with velocity vector perpendicular to position vector? Can you say anything about it at all?
I know that velocity is always perpendicular to the position vector for circular motion and at the endpoints of elliptical motion. Is there a general statement that can be made about the object's motion when the velocity is perpendicular to position?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

regulerenes4w
Answered 2022-04-19 Author has 10 answers

If you are constrained to move in a plane, then yes, circular motion is the only possibility. To see this, let (x(t),y(t)) be the coordinates at time t on the path. The condition is
x dx  dt +y dy  dt =0
Integrate with respect to t:
xdxdtdt+ydydtdt=0,dt

x22+y22=C
which is the equation of a circle.
So your path has to move around the circle. But that doesn't mean the path itself is a circle. It can speed up. It can slow down. It can stop and reverse direction, all without violating the perpendicularity condition (when it is stopped, the velocity is 0, whose inner product with the position vector is still 0 - this stretches the definition of "perpendicular" a bit, but generally it's allowed, as the physical objects being modelled occasionally do stop).
In three dimensions, a similar calculation shows you that the motion is constrained to some sphere. But you can move in all sorts of ways on a sphere that do not qualify as "circular".

Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-07-01
Can you apply the fundamental theorem of calculus with the variable inside the integrand?
For example: d d x ( a x x f ( t ) d t )
asked 2022-02-10
How do you find the eccentricity, directrix, focus and classify the conic section r=21+2cosθ?
asked 2022-06-26
How to calculate the integral 0.5 1 1 2 x x 2 ?
asked 2021-08-06
The analysis of tooth shrinkage by C. Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process has not yet come to a halt. In northern Europeans, for example, tooth size reduction now has a rate of 1% per 1000 years.
a. If t represents time in years and y represents tooth size, use the condition that y=0.99y0 when t = 1000 to find the value of k in the equation y=y0ekt. Then use this value of k to answer the following questions.
b. In about how many years will human teeth be 90% of their present size?
c. What will be our descendants’ tooth size 20,000 years from now (as a percentage of our present tooth size)?
asked 2021-08-06
Evaluate the indefinite integral.
(4x3)2dx
asked 2022-02-12
L1[s(s+3)2+9]
asked 2022-02-13
Properties of dual linear transformation
Let V,W finite dimensional vector spaces over the field F, and let T:VW a linear transformation.
Define T:WV the dual linear transformation. i.e T(ψ)=ψT.

New questions