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(

Question

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.

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Hồng Cúc 6 months 2021-08-10T15:13:17+00:00 2 Answers 16 views 0

Answers ( )

    0
    2021-08-10T15:15:07+00:00

    Answer:

    Explanation:

    Given

    x(t) = r cos(ωt)

    y(t) = r sin(ωt)

    where r = 3.84×10⁸ m and ω = 2.46×10-⁶radians/s

    Average velocity = Δx /Δt

    At t = 0s

    x(0) = 3.84×10⁸×cos(2.46×10-⁶×0)

    x(0) = 3.84×10⁸×1 = 3.84×10⁸m

    At t = 3.92days = 3.92×86400s = 3.39×10⁵s

    x(3.39×10⁵) = 3.84×10⁸×cos(2.46×10-⁶×3.39×10⁵) = –2.58×10⁸m

    Δx = x(3.39×10⁵) – x(0) = –2.58×10⁸–3.84×10⁸m = –6.42×10⁸m

    Δt = 3.39×10⁵ – 0 = 3.39×10⁵

    vx = –6.42×10⁸/3.39×10⁵ = –1894m/s

    At t = 0s

    y(0) = 3.84×10⁸×sin(2.46×10-⁶×0)

    y(0) = 3.84×10⁸×0= 0m

    At t = 3.92days = 3.92×86400s = 3.39×10⁵s

    y(3.39×10⁵) = 3.84×10⁸×sin(2.46×10-⁶×3.39×10⁵) = 2.84×10⁸m

    Δy = y(3.39×10⁵) – y(0) = 2.84×10⁸ – 0 = 2.84×10⁸m

    Δt = 3.39×10⁵ – 0 = 3.39×10⁵

    vy = 2.84×10⁸/3.39×10⁵= 838m/s

    vy = 838m/s

    v = √(vx² + vy²) = √((-1894)² + 838²)

    v = 2071m/s

    θ = tan-¹(vy/vx) = tan-¹(838/1894) = -23.9°

    0
    2021-08-10T15:15:13+00:00

    Answer: 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°

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