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8 December, 02:21

A Federation starship (8.5 ✕ 106 kg) uses its tractor beam to pull a shuttlecraft (1.0 ✕ 104 kg) aboard from a distance of 14 km away. The tractor beam exerts a constant force of 4.0 ✕104 N on the shuttlecraft. Both spacecraft are initially at rest. How far does the starship move as it pulls the shuttlecraft aboard?

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  1. 8 December, 02:32
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    Refer to the diagram shown below.

    m₁ = 8.5 x 10⁶ kg, the mass of the starship

    m₂ = 10⁴ kg, the mass of the shuttlecraft

    a₁ = the acceleration of the starship

    a₂ = the acceleration of the shuttle

    F = 4 x 10⁴ N, the pulling force

    Let y = distance traveled by the starship

    Let x = distance traveled by the shuttlecraft

    If t = the time of travel, then

    y = 0.5a₁t² (1)

    x = 0.5a₂t² (2)

    F = m₁a₁ = m₂a₂ (3)

    Also,

    x + y = 14000 m (4)

    From (2), obtain

    a₁ = (4 x 10⁴ N) / (8.5 x 10⁶ kg) = 4.706 x 10⁻³ m/s²

    a₂ = (4 x 10⁴ N) / (10⁴ kg) = 4 m/s²

    From (1), (2) and (4), obtain

    0.5 * (t s) ² * (4 + 4.706 x 10⁻³ m/s²) = 14000 m

    2.002353t² = 14000

    t² = 6991.774 s²

    t = 83.617 s

    Therefore

    x = 0.5*4*6991.774 = 13984 m = 13.984 km

    y = 0.5*4.706 x 10⁻³*6991.774 = 16.452 m

    The starship moves only about 16.5 m while pulling in the shuttlecraft by 13.98 km.

    Answer: The starship moves by 16.5 m (nearest tenth)
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