Finding Equations of Movement and Acceleration Along a Parabola

In summary, the conversation discusses a problem involving a point moving along a parabola with a given equation and velocity. The goal is to find the equation of movement, radial acceleration, and transverse acceleration. The individual has attempted to solve the problem by rewriting the equation of the parabola and using differentiation, but is unsure if their approach is correct. They are seeking guidance and mention having knowledge of differential equations.
  • #1
CCMarie
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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.
 
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  • #2
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?
 
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  • #3
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?
 
  • #4
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##.
 
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  • #5
Thank you very much, Delta^2!
 

Related to Finding Equations of Movement and Acceleration Along a Parabola

What is movement along a parabola?

Movement along a parabola refers to the motion of an object following a curved path that is defined by a parabolic shape. This type of movement is often seen in projectile motion, where an object is launched at an angle and follows a parabolic path until it reaches the ground.

What factors affect movement along a parabola?

The factors that affect movement along a parabola include the initial velocity of the object, the angle at which it is launched, the force of gravity, and any external forces acting on the object, such as air resistance.

How is the velocity of an object determined during movement along a parabola?

The velocity of an object during movement along a parabola can be determined using the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration due to gravity, and t is the time elapsed.

What is the difference between movement along a parabola and linear motion?

Movement along a parabola is characterized by a curved path, while linear motion follows a straight path. Additionally, the acceleration of an object in linear motion remains constant, while the acceleration in movement along a parabola changes over time due to the force of gravity.

How is the height of an object at a specific point along a parabola calculated?

The height of an object at a specific point along a parabola can be calculated using the equation h = ut + 1/2at^2, where h is the height, u is the initial velocity, a is the acceleration due to gravity, and t is the time elapsed.

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