Finding maximum and minimum values of vel. and acc. of a particle on an ellipse

In summary, the problem asks to find the maximum and minimum values of the magnitude of velocity and acceleration for a particle moving in an ellipse in the yz-plane. The provided equations for velocity and acceleration are used to find the expressions for the magnitude of velocity and acceleration. The concept of local maxima and minima is discussed, and it is determined that the first derivative of a function at a local max or min is 0. The process of finding the local max and min values is explained, and it is recommended to use either the second derivative or plugging values into the original equations to determine if the values are max or min. The solution to the problem is found by plugging in the values given and checking for errors. The ask
  • #1
JoeSabs
9
0

Homework Statement


A particle moves around the ellipse ((y/3)^2)+((z/2)^2)=1 in the yz-plane in such a way that its position at time t is r(t)=(3cost)j+(2sint)k. Find the maximum and minimum values of |v| and |a|. (Hint: Find the extreme values of |v|^2 and |a|^2 first and take square roots later.)


Homework Equations


v(t)= (-3sint)j+(2cost)k
a(t)= (-3cost)j+(-2sint)k
|v|= sqrt(((-3sint)^2)+((2cost)^2))
|a|= sqrt(((-3cost)^2)+((-2sint)^2))

The Attempt at a Solution


I got those equations, but our teacher never showed us how to find the extrema of these equations. Coming out of a horrible calc II class, I'm not exactly sure how to evaluate these. Our current teacher has a bad habit of teaching the class after the homework's due...
 
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  • #2
What do you know about local minimums and maximums of a function? What is the derivative of a function at a local max or min? v(t) and a(t) are both periodic functions. do periodic functions have any absolute maximums or minimums?
 
  • #3
Right, so the derivative of a local max/min is 0. does that mean i have to find the third derivative to find the max/mins for a(t)? As for periodic functions, they repeat, so they have a max and min that repeats. I'm not asked for the absolute max/min, so I'm assuming they want the relative.

so after plugging zero in for a(t) (to find relatives for v(t)) and a'(t) (to find relatives for a(t)) I got 3 and 2, respectively. I checked the back, and it states that each has a max and min of 3 and 2, respectively. What am I missing that I'm only getting half of the answer?
 
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  • #4
Periodic functions won't have absolute max/mins but will have local max/mins. At a local max or min, the 1st derivative vanishes. At a local max, the second derivative is negative, and at a local min, the second derivative is positive.

So, just find out where the 1st derivatives of v(t) and a(t) are zero, then find out whether those values are maximums or minimums either by taking the second derivatives and determining whether they're pos or neg there or plugging the values of t into v(t) and a(t) and seeing when you get a larger v(t) or a(t) and when you get a smaller v(t) and a(t).
 
  • #5
I'm working backwards at the moment, plugging the answers given into the problem. They don't give derivatives which equal zero.EDIT:

Figured it out. Thanks!

My Prof Said I was wrong :( Can someone show me the work for this problem? I'm not sure but I think I may have made a sign error.
 
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  • #6
Show what you did and what answer you got. I'm a bit confused about why you "coming out of a horrible Calculus II course" would affect this problem. Most people learn to find max and min in Calculus I. It certainly shouldn't be the responsibility of your Calculus III teacher to show you how to find max and min of a function of one variable.
 

FAQ: Finding maximum and minimum values of vel. and acc. of a particle on an ellipse

What is an ellipse?

An ellipse is a geometric shape that resembles a flattened circle. It is defined as the set of all points in a plane, the sum of whose distances from two fixed points (called the foci) is constant.

How do you find the maximum and minimum values of velocity and acceleration of a particle on an ellipse?

To find the maximum and minimum values of velocity and acceleration of a particle on an ellipse, you can use the parametric equations of motion and the derivatives of these equations. Alternatively, you can use the geometric properties of the ellipse, such as the semi-major and semi-minor axes, to determine the maximum and minimum values.

What is the relationship between velocity and acceleration on an ellipse?

The relationship between velocity and acceleration on an ellipse can be described by the parametric equations of motion. The velocity of a particle on an ellipse is tangent to the curve at any point, while the acceleration is the rate of change of the velocity. This means that the acceleration is perpendicular to the velocity at any given point on the ellipse.

How does the shape of the ellipse affect the maximum and minimum values of velocity and acceleration?

The shape of the ellipse can greatly affect the maximum and minimum values of velocity and acceleration. For example, an elongated ellipse will result in higher maximum values of velocity and acceleration, while a more circular ellipse will have lower maximum values. Additionally, the orientation of the ellipse can also impact the maximum and minimum values.

What are some real-life applications of finding maximum and minimum values of velocity and acceleration on an ellipse?

Finding maximum and minimum values of velocity and acceleration on an ellipse can be useful in various fields, such as physics, engineering, and astronomy. For example, in physics, this can help in understanding the motion of planets and satellites in elliptical orbits. In engineering, it can be used to optimize the performance of machines with circular or elliptical components. In astronomy, it can aid in predicting the trajectories of comets and other celestial bodies that move in elliptical orbits.

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