Finding Equations of Movement and Acceleration Along a Parabola

[CCMarie]

I have this problem to solve until tomorrow:

A point moves along the parabola r*cos^2(θ/2) = p/2, p > 0, in the direction that θ increases. At the time t=0, the point is on the verge of the parabola. The velocity is v = k*r, k>0.
What is the equation of movement, the radial acceleration and the transverse acceleration.

I've tried to solve it by rewriting the equation of the parabola r = (-2d)/(1+cosθ), and than I thought about derivation of this equation as both r and theta are functions of t, r(t) and θ(t).

I tried to find the acceleration that is the first derivative of the velocity a = k r'(t). And I wrote the equation of movement as s(t) = 1/2*k*t^2*r'(t).

I don't know if what I did so far is ok. And I am not quite sure how to work with the radial and transverse acceleration.

I would really appreciate some clues. Thank you.

 

[Delta2]

I am not sure what you mean by equation of movement (equation of the distance traveled as a function of time t? then your equation for s(t) may not be correct if r'(t) is not constant in time)

However the equation your write for acceleration is the magnitude of transverse acceleration $a_t(t)=kr'(t)$. The magnitude of radial acceleration is $a_r(t)=v^2(t)/r(t)=k^2r(t)$.

I believe the assignment asks to write down and maybe solve the differential equations for $r(t)$ and $\theta(t)$. Have you done any differential equations in your class?

 

[CCMarie]

Thank you for your response, [Delta][/2].

By the equation of movement I meant the distance traveled as a function of time. In my case I guess that r(t) shouldn't be constant in time. Am I going to have a partial differential in the expression of s(t)?

I have studied differential equations. But I don't know what are the equations that I have to solve? Where do I obtain the differential equation to be solved from?

 

[Delta2]

Then the equation of movement is $s(t)=\int v(t)dt+c=\int kr(t)dt+c$

At the moment I am abit sleepy and can't think if we got enough data to find $r(t)$ or $\theta(t)$ as function of time. Of course if we find one of them then we know the other cause they are connected with the equation $r(t)cos^2(\theta(t)/2)=p/2$.

 

[CCMarie]

Thank you very much, Delta^2!