Centripetal Acceleration of rocket car

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Homework Statement


A rocket car is constrained to move on an elliptical track (semi-major axis [itex] a [/itex] and semi-minor axis [itex] b [/itex]). The car is moving at a constant speed [itex] v_0 [/itex]. Determine the acceleration of the car in [itex] \frac{m}{s^2} [/itex]. [itex] a = 4 \hspace{2 mm} km [/itex], [itex] b = 2 \hspace{2 mm} km [/itex], and [itex] v_0 = 360 \frac{km}{hr} [/itex].

Homework Equations




The Attempt at a Solution


My thought for this problem was to try and find the value of 'R' using the definition of curvature: [itex] \kappa = \frac{1}{R} [/itex]. Using the definition of an ellipse: [itex] \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 [/itex], I solved for 'y' and ended up with: [itex] y = b \sqrt{1- \frac{x^2}{a^2}}[/itex]. Then, I used the definition of curvature [itex] \kappa = \frac{\frac{d^2y}{dx^2}}{(1+(\frac{dy}{dx})^2)^\frac{3}{2}} [/itex]. Then, I took the first and second derivative of my equation for an ellipse (I will not write them here because it gets very messy). Then, I substituted my second derivative into the equation for curvature. Finally, using the fact that [itex] \kappa = \frac{1}{R} [/itex], therefore [itex] R = \frac{1}{\kappa} [/itex]. So I substituted my equation for R (1 over kappa) into the equation for centripetal acceleration( [itex] a = \frac{v^2}{R} [/itex] and used the given [itex] v_0 = 360 \frac{km}{hr} [/itex]. However, as you can probably guess, this is extremely messy. Furthermore, I do not see how I am to get rid of the x and y in my equations (the answer is meant to be a numeric value). So I am certain I have done something wrong but I'm not sure what...
 

Answers and Replies

  • #2
Simon Bridge
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What doe "R" represent? You have to say... is the idea to use the equation ##a=v^2/R##?
Note: the acceleration of the car won't be a constant since it's rate of change in direction varies but it's speed doesn't.
 
  • #3
gneill
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You might get neater expressions for the first and second derivatives if you use implicit differentiation.
 
  • #4
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What doe "R" represent? You have to say... is the idea to use the equation ##a=v^2/R##?
Note: the acceleration of the car won't be a constant since it's rate of change in direction varies but it's speed doesn't.
Sorry, I realize I wasn't very clear. The idea was to use the equation [itex] a = \frac{v^2}{R} [/itex].. I attempted to find R using the equation [itex] \kappa = \frac{1}{R} [/itex]. So I solved for R using the curvature equation and then plugged this value into the equation for centripetal acceleration but it is just so messy that I think I probably am not on the right track. I hope this clears things up...
 
  • #5
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You might get neater expressions for the first and second derivatives if you use implicit differentiation.
I hadn't thought about that but I'll certainly give it a shot, thanks :D
 
  • #6
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Note: the acceleration of the car won't be a constant since it's rate of change in direction varies but it's speed doesn't.
So in that case I am basically at the solution, I just need to clean things up algebraically I think. I have this bad habit where if my solution looks really messy I just assume I've done something wrong..
 
  • #7
Simon Bridge
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I can't tell, because you didn't answer the questions.
 
  • #8
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I can't tell, because you didn't answer the questions.
Which questions are you referring to? R is meant to represent radius of curvature, is there something wrong with how I've approached the problem?
 

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