Question

If the hula hoop is positioned vertically on the floor so that the black band is touching the floor, when the hula hoop is given a push, it begins to travel upright along the floor. The path traced out by the black band is given by the position vector r= (t – sin t) i + (1 – cos t) j.

c) Find the maximum and minimum values of the velocity vector analytically.

d) Find the maximum and minimum values of the acceleration vector analytically.

e) Provide a graphical representation of the velocity and acceleration vectors. Do they confirm your answers to c and d?

Answer 1

d)

For finding the maxima and minima, we can take the modulus

The derivative will be given by the formula

Hence the maxima will occur, when sin(t) = 0, which implies the point cos(t) = n*pi

The maximum velocity will be given when cos(t) = -1, which implies the maximum velocity is given by 2i and the magnitude of the velocity will be 2

The minimum will be given when cos(t) = 1, hence the magnitude will be 0 in this case

e)

The maxima will occur, when sin(t) = cos(t), which implies t = pi/4

Since at that point sin(t) = cos(t) = 1/sqrt(2)

Hence, the correct answer will be a(t) = 1/sqrt(2) i + 1/sqrt(2) j

f) The graphical representation of the vectors will be

(Blue curve)

(Red curve)

As per the parametric plot, the blue curve goes to have a maximum value of x being 2 and the value of maximum being equal to 1

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