"An ant is crawling on a horizontal flat table's surface. In polar coordinates, his coordinates are r(t) = bt^2 and theta(t) = (omega)t where b and omega are known constants.
a) Find his velocity function v(t) in terms of t, b, omega and the polar unit vectors r-hat and theta-hat.
b) Find his acceleration vector a(t) in terms of t, b, omega and the polar units r-hat and theta-hat.
c) What is his speed when t = 1/(omega)?
d) How far is he from the origin when t = 1/(omega)?"
Could someone help me with this problem, please? I'd appreciate it!
a) Find his velocity function v(t) in terms of t, b, omega and the polar unit vectors r-hat and theta-hat.
b) Find his acceleration vector a(t) in terms of t, b, omega and the polar units r-hat and theta-hat.
c) What is his speed when t = 1/(omega)?
d) How far is he from the origin when t = 1/(omega)?"
Could someone help me with this problem, please? I'd appreciate it!
-
look in your book to find that the expression for velocity in polar coords is
v(t)= r dot r hat + r theta dot theta hat
where dot means derivative with respect to time
therefore, v(t)=2bt r hat + bt^2* omega theta hat
the expression for acceleration is a bit more complex
a(t)=(r double dot - r theta dot squared) r hat + (2 r dot theta dot + r theta double dot)theta hat
where double dot means the second time derivative
use the expressions for r(t) and theta(t) and solve for acceleration
v(t)= r dot r hat + r theta dot theta hat
where dot means derivative with respect to time
therefore, v(t)=2bt r hat + bt^2* omega theta hat
the expression for acceleration is a bit more complex
a(t)=(r double dot - r theta dot squared) r hat + (2 r dot theta dot + r theta double dot)theta hat
where double dot means the second time derivative
use the expressions for r(t) and theta(t) and solve for acceleration