Question

Position and velocity of a point are given in polar coordinates
by *R* = 2, θ = 35 degrees, and
**v** = 4**R** + 3Θ. The 35
degrees is measured positive counterclockwise from the
*x*-axis on an *xy* Cartesian coordinate frame. What
is the velocity of the point in terms of **i** and
**j**?

Answer #1

here x^ and y^ is i^ and j^ respectively

Given any Cartesian coordinates, (x,y), there are polar
coordinates (?,?)(r,θ) with −?2<?≤?2.−π2<θ≤π2.
Find polar coordinates with −?2<?≤?2−π2<θ≤π2 for the
following Cartesian coordinates:
(a) If (?,?)=(18,−10)(x,y)=(18,−10) then
(?,?)=((r,θ)=( , )),
(b) If (?,?)=(7,8)(x,y)=(7,8) then
(?,?)=((r,θ)=( , )),
(c) If (?,?)=(−10,6)(x,y)=(−10,6) then
(?,?)=((r,θ)=( , )),
(d) If (?,?)=(17,3)(x,y)=(17,3) then
(?,?)=((r,θ)=( , )),
(e) If (?,?)=(−7,−5)(x,y)=(−7,−5) then
(?,?)=((r,θ)=( , )),
(f) If (?,?)=(0,−1)(x,y)=(0,−1) then (?,?)=((r,θ)=( ,))

The Cartesian coordinates of a point are given. (a) (−3, 3)
(i) Find polar coordinates (r, θ) of the point, where r > 0
and 0 ≤ θ < 2π.
(r, θ) =
(ii) Find polar coordinates (r, θ) of the point, where r < 0
and 0 ≤ θ < 2π.
(r, θ) =
(b) (4, 4 sq root3 ) (i) Find polar coordinates (r, θ) of the
point, where r > 0 and 0 ≤ θ < 2π....

The Cartesian coordinates of a point are given. (a) (−4, 4) (i)
Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤
θ < 2π. (r, θ) (ii) Find polar coordinates (r, θ) of the point,
where r < 0 and 0 ≤ θ < 2π. (r, θ) (b) (3, 3 3 ) (i) Find
polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ
< 2π. (r, θ) =...

The Cartesian coordinates of a point are given.
(a) (5
3
, 5)(i) Find polar coordinates (r, θ) of the point,
where
r > 0 and 0 ≤ θ < 2π.
(r, θ) =
(ii) Find polar coordinates (r, θ) of the point, where
r < 0 and 0 ≤ θ < 2π.
(r, θ) =
(b)
(1, −3)
(i) Find polar coordinates (r, θ) of the point,
where
r > 0 and 0 ≤ θ <...

The Cartesian coordinates of a point are given.
(a) (−8, 8)
(i) Find polar coordinates
(r, θ) of the point, where
r > 0 and 0 ≤ θ < 2π.
(ii) Find polar coordinates
(r, θ) of the point, where
r < 0 and 0 ≤ θ < 2π.
(b) (4,4sqrt(3))
(i) Find polar coordinates
(r, θ) of the point, where
r > 0 and 0 ≤ θ < 2π.
(ii) Find polar coordinates (r, θ)...

given the polar curve r = 2(1+cos theta) find the Cartesian
coordinates (x,y) of the point of the curve when theta = pi/2 and
find the slope of the tangent line to this polar curve at theta =
pi/2

1. You are given the point P in the cartesian coordinates (−4,
−4). Write the point in polar coordinates given the
restrictions:
(a) r > 0, and 0 ≤ θ < 2π. (in these programs, r = 0 is
just defined to be the origin).
(b) r<0andθ∈[0,2π)
(c) Write the point in polar coordinates that represent the
same point P but that
is different than the previous parts.

Your task will be to derive the equations describing the
velocity and acceleration in a polar coordinate
system and a rotating polar vector basis for an object in general
2D motion starting from a general
position vector. Then use these expressions to simplify to the case
of non-uniform circular motion, and
finally uniform circular motion.
Here's the time-dependent position vector in a Cartesian coordinate
system with a Cartesian vector
basis: ⃗r(t)=x (t)
̂
i+y(t)
̂
j where x(t) and y(t)...

Find the velocity and position vectors of a particle that has
the given acceleration and the given initial velocity and position.
a(t) = (6t + et) i + 12t2 j, v(0) = 3i, r(0) = 7 i − 3 j
v(t)=
r(t)=

Find the velocity and position vectors of a particle that has
the given acceleration and the given initial velocity and
position.
a(t) = 2 i +
6t j + 12t2
k, v(0) = i,
r(0) = 3 j − 6
k

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