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 the equations x = 4.33 sin t and y = 4.00 4.33 cos t, where is a constant, x and y are in meters, and t is in seconds. (a) Determine the components of velocity of the object at t = 0. (Use the following as necessary: .) v with arrow = m/s (b) Determine the components of the acceleration of the object at t = 0. (Use the following as necessary: .) a with arrow = m/s2 (c) Write expressions for the position vector, the velocity vector, and the acceleration vector of the object at any time t > 0. (Use the following as necessary: omega for and t.) r with arrow = m v with arrow = m a with arrow = m (d) Describe the path of the object in an xy plot.

Explanation / Answer

a. velocity is time rate of distance

so

Vx = dx/dt = -4.33 sin wt /dt = - 4.33 w cos wt

Vy = dy/dt = (4- 4.33 cos wt)/dt = +4.33 w sin wt

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accelration is time rate of velocity

so

ax = dVx/dt = -4.33 W cos wt/dt = +4.33 W^2 sin wt

ay = dVy/dt = 4.33 W sin Wt/dt = -4.33 W^2 cos wt

c. position vector r = -4.33 sin wt i + (4 - 4.33 cos wt) j

velocity vector v = -4.33 cos wti + 4.33 sin wt

accleration vector a = 4.33 w^2 sin wt i - 4.33 w^2 cos wt

d, this follows a sinusoidal wae form

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