Speed in radial coordinates and angular coordinates

In summary, the question is asking for the speed at an arbitrary time, and using the equation for velocity, calculates the speed.
  • #1
mrchauncey
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Homework Statement



First off, I am not a physics student. I am a math major taking a maple software course and there is a question that I can not figure out.

The question gives me a radial coordinates r

r:= [itex]\frac{a*t^{2}*e^{-b*t}}{1+t^{2}}[/itex]

And angular coordinates:

θ:=b+c*t[itex]^{2/3}[/itex]

Where a,b,c are real constants.

It then puts the radial and angular coordinates in the form

X:= r*cos(θ)
Y:=r*sin(θ) so it looks like this:

X:=[itex]\frac{a*t^2*e^{-b*t}*cos(b+c*t^{2/3}}{1+t^{2}}[/itex]

Y:=[itex]\frac{a*t^{2}*e^{-b*t}*sin(b+c*t^{2/3}}{1+t^{2}}[/itex]

Which I understand.

Homework Equations



Now the question is calculate speed V at an arbitrary time simplifying as much as possible. This is where I get confused. The question then says to find speed V at time t take

u=([itex]\frac{dX}{dt}[/itex])[itex]^{2}[/itex]+([itex]\frac{dY}{dt}[/itex])[itex]^{2}[/itex]

[itex]\sqrt{u}[/itex]

It does not explain why they use that formula to calculate the speed. Just wondering if anyone can shed some light on this situation. This is an example in the book so I know this is how you do it or the way they want me to do it, so I am not looking for an answer, just an explanation.

The Attempt at a Solution



Thanks for looking.
 
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  • #2
mrchauncey said:
The question then says to find speed V at time t take

u=([itex]\frac{dX}{dt}[/itex])[itex]^{2}[/itex]+([itex]\frac{dY}{dt}[/itex])[itex]^{2}[/itex]

[itex]\sqrt{u}[/itex]

It does not explain why they use that formula to calculate the speed.
Velocity is defined at the rate of change of position, which in one dimension gives
$$
v = \frac{dx}{dt}
$$
In 2D, as for you problem, velocity is a vector, ##\vec{v} = (v_x, v_y)^T##, with
$$
v_x = \frac{dx}{dt}, \quad v_y = \frac{dy}{dt}
$$

Speed is simply the magnitude of the velocity (L2-norm), hence the formula you were given.
 
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  • #3
Thanks. I actually just found that reading through my old calculus textbook on parametric equations, arc length, and speed. Its been a while since I took calculus and was never the best at speed, velocity and acceleration, etc. My math background has more to do with Actuarial Science and Stats. Once again thanks again. Have a nice day.
 

Related to Speed in radial coordinates and angular coordinates

1. What is the difference between speed in radial coordinates and angular coordinates?

Speed in radial coordinates refers to the speed of an object moving in a straight line, while angular coordinates refer to the speed of an object moving in a circular path around a fixed point. In other words, radial coordinates measure the distance traveled by an object from its starting point, while angular coordinates measure the change in the direction of the object's motion.

2. How is speed calculated in radial coordinates?

Speed in radial coordinates is calculated using the formula v = r/t, where v is the speed, r is the distance traveled, and t is the time taken.

3. What is the unit of measurement for speed in angular coordinates?

Speed in angular coordinates is measured in radians per second (rad/s). Radians are a unit of measurement for angles, and 1 radian is equal to the angle formed by an arc of a circle with a length equal to the radius of the circle.

4. Can an object have different speeds in radial and angular coordinates at the same time?

Yes, an object can have different speeds in radial and angular coordinates at the same time. For example, a car moving in a circular path at a constant speed will have a constant angular speed but varying radial speed as it moves closer or farther away from the center of the circle.

5. How does the speed in radial and angular coordinates affect the overall motion of an object?

The speed in radial and angular coordinates determines the overall motion of an object. A combination of radial and angular speeds can result in various types of motion, such as circular, elliptical, or straight-line motion.

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