The Moon orbits the Earth in an approximately circular path. The position of the moon as a function of time is given by: x(t) = r cos(ωt) y(t) = r sin(ωt) where r = 3.84 108 m and ω = 2.46 10-6 radians/s. What is the average velocity of the Moon measured over the interval from t = 0 to t = 3.92 days? Find its magnitude and direction, given as an angle measured counterclockwise from the positive x-axis.

Average Velocity = - 643.42 i + 512.66 j m/s

Magnitude = 822.7 m/s

Direction = 141.45°

Explanation:

r = 3.84 x 10^8 m

w = 2.46 x 10^-6 rad/s

Formula for Average velocity = displacement / time

at t = 0

x(0) = r

y(0) = 0

at t = 8.45 days

= 8.45 x 24 x 3600 s =730080 sec

w t = 2.46 x 10^-6 x 730080 = 1.80 rad Or 102.90°

xf = r cos(w t) = - 0.2233r

yf = r sin(w t) = 0.9747r

Displacement = (xf - x0)i + (yf - y0)j = -1.2233r i + 0.9747r j

<v> = dispalcement / t = (-1.2233r i + 0.9747r j ) / (730080 s )

= - 643.42 i + 512.66 j m/s

Magnitude

= sqrt(643.42^2 + 512.66^2)

= 822.7 m/s

Direction

= 180 - tan^-1(512.66 / 643.42)

= 141.45°