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8 February, 19:08

Mt. Asama, Japan, is an active volcano. In 2009, an eruption threw solid volcanic rocks that landed 1 km horizontally from the crater. If the volcanic rocks were launched at an angle of 40° with respect to the horizontal and landed 900 m below the crater, (a) what would be their initial velocity and (b) what is their time of flight?

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  1. 8 February, 19:15
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    a) 69.3 m/s

    b) 18.84 s

    Explanation:

    Let the initial velocity = u

    The vertical and horizontal components of the velocity is given by uᵧ and uₓ respectively

    uᵧ = u sin 40° = 0.6428 u

    uₓ = u cos 40° = 0.766 u

    We're given that the horizontal distance travelled by the projectile rock (Range) = 1 km = 1000 m

    The range of a projectile motion is given as

    R = uₓt

    where t = total time of flight

    1000 = 0.766 ut

    ut = 1305.5

    The vertical distance travelled by the projectile rocks,

    y = uᵧ t - (1/2) gt²

    y = - 900 m (900 m below the crater's level)

    -900 = 0.6428 ut - 4.9t²

    Recall, ut = 1305.5

    -900 = 0.6428 (1305.5) - 4.9 t²

    4.9t² = 839.1754 + 900

    4.9t² = 1739.1754

    t = 18.84 s

    Recall again, ut = 1305.5

    u = 1305.5/18.84 = 69.3 m/s
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