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Math Help - 2 gravitational problems

  1. #1
    Junior Member winterwyrm's Avatar
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    2 gravitational problems

    A projectile is shot directly away from Earth's surface. Neglect the rotation of the Earth. What multiple of Earth's radius RE gives the radial distance (from the Earth's center) the projectile reaches if (a) its initial speed is 0.672 of the escape speed from Earth and (b) its initial kinetic energy is 0.672 of the kinetic energy required to escape Earth? (Give your answers as unitless numbers.)

    and

    Mountain pull. A large mountain can slightly affect the direction of “down” as determined by a plumb line. Assume that we can model a mountain as a sphere of radius R = 3.00 km and density (mass per unit volume) 2.6 × 103 kg/m3. Assume also that we hang a 0.250 m plumb line at a distance of 3R from the sphere's center and such that the sphere pulls horizontally on the lower end. How far would the lower end move toward the sphere?
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  2. #2
    MHF Contributor
    skeeter's Avatar
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    let k = 0.672

    v_0 = k\sqrt{\frac{2GM_e}{R_e}}

    E_0 = \frac{1}{2}mv_0^2 - \frac{GM_em}{R_e}

    E_f = -\frac{GM_em}{nR_e}

    set E_0 = E_f and solve for n

    for part (b) ...

    v = \sqrt{\frac{2GM_e}{R_e}}

    E_0 = \frac{k}{2}mv^2 - \frac{GM_em}{R_e}

    same E_f ... same procedure, solve for n.
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