The coordinates of an object moving in the xy plane vary with time according to

Question

The coordinates of an object moving in the xy plane vary with time according to x= -(5.00 m) sin(wt) and y= (4.00 m) - (5.00 m)cos(wt), where w is a constant and t is in seconds.
a) Determine the components of velocity and components of accelerationof the object at t=0.
b) Write expressions for the position vector, the velocity vector, and the acceleration vector of the object at any time t>0.
c) Descibe the path of the object in an xy plot.

Explanation / Answer

a) First take the derivative of the position function with respect to t which gives you: v_x = -(5.00 m)(w)cos(wt) v_y = (5.00 m)(w)sin(wt) The above are your components of velocity and when t = 0: v_x = -(5.00 m)(w) v_y = 0 Taking the derivative of the velocity function and plugging in t = 0 you get: a_x = (5.00 m)(w^2)sin(wt) = 0 a_y = (5.00 m)(w^2)cos(wt) = (5.00 m)(w^2) b) Being allowed to choose any time t>0 you typically choose t=1, as it often results in the easiest calculations. Plugging in t=1 to the given formulas and the ones we solved above you get: x = -(5.00 m)sin(w) y = (4.00 m) - (5.00 m)cos(w) v_x = -(5.00 m)(w)cos(w) v_y = (5.00 m)(w)sin(w) a_x = (5.00 m)(w^2)sin(w) v_y = (5.00 m)(w^2)cos(w) c) As for the description of the path: The path is clockwise around a circle of radius 5 m centered at (0,4) m, starting at the lowermost point on the circle. Hope this helps.

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