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aeb2335
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Homework Statement
I have elected to put this in the "math" section because it is primarily a math question however, please read to problem knowing that it came from an engineering dynamics textbook.
Given:
(as written)
A particle moving along y= x - (x^2 / 400) where x and y are in ft. If the velocity component in the x direction remains constant at Vx= 2 ft/s , determine the magnitudes of velocity and acceleration when x = 20 ft.
Homework Equations
V = ds/dt
A = dv/dt
The Attempt at a Solution
Ok, at first I became a little confused so I drew a Cartesian co-ordinate system and plotted the curve Y going up X horizontal. I then realized that its a path equation so it needs to be differentiated.
The problem becomes I don't really understand the notation
It should possible differentiate the path saying f(x,y)'= the velocity function
What I don't get is how you can use Leibniz's notation for this problem to solve it. After being frustrated at the lack of clarity I looked through the book and at the solution and it appears the book's notes on how to solve this problem and the solution uses Newtonian notation.
I apologize for not knowing how to make the dot character
Saying that (first line of solution)
ydot = xdot - (2x (xdot) / 400 )
The problem is that I tend to drop terms like the underlined xdot above because I am not really sure where it comes from. It seems to me that there is some implicit magic going on. I would have no problem in Leibniz notation and the reason this problem gave me pause was because it didn't appear to be easy to express in Leibinz notation.
Can the question be properly framed by saying given a curve S(x,y) such that
S(x,y)= x-(x^2/400)-y
then saying:
S(x,y)ds/dt=(x-(x^2/400)-y)ds/dt
V(x,y) = x ds/dt - (x^2 ds/dt / 400) - y ds/dt ?
would it be even better to say
S(x,y)= x(i) -(x^2(i)/400)-y(j) ?
Is there a guide or reference to Newtonian notation (other that wikipedia etc.) that makes it easy to understand?
Does Newtonian notation in this example treat X and Y as functions themselves and not variables which is why the x dot term exists? And if so would X and Y Always notate a function and directions would be explicitly in i,j,k ? Furthermore why would you treat X and Y as a functions in this equation when it also corresponds directly to the inputs of that function?
Also is there a guide to uber correct and uniform notation and/or an explanation of the mathematical and physical symbols.
Thanks so much