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24 May, 17:03

An ice sled powered by a rocket engine starts from rest on a large frozen lake and accelerates at + 44 ft/s2. After some time t1, the rocket engine is shut down and the sled moves with constant velocity v for a time t2. If the total distance traveled by the sled is 18,350 ft and the total time is 90 s, find the following. (a) the times t1 and t2 t1 = s t2 = s (b) the velocity v

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  1. 24 May, 20:55
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    a) t₁ = 4.76 s, t₂ = 85.2 s

    b) v = 209 ft/s

    Explanation:

    Constant acceleration equations:

    x = x₀ + v₀ t + ½ at²

    v = at + v₀

    where x is final position,

    x₀ is initial position,

    v₀ is initial velocity,

    a is acceleration,

    and t is time.

    When the engine is on and the sled is accelerating:

    x₀ = 0 ft

    v₀ = 0 ft/s

    a = 44 ft/s²

    t = t₁

    So:

    x = 22 t₁²

    v = 44 t₁

    When the engine is off and the sled is coasting:

    x = 18350 ft

    x₀ = 22 t₁²

    v₀ = 44 t₁

    a = 0 ft/s²

    t = t₂

    So:

    18350 = 22 t₁² + (44 t₁) t₂

    Given that t₁ + t₂ = 90:

    18350 = 22 t₁² + (44 t₁) (90 - t₁)

    Now we can solve for t₁:

    18350 = 22 t₁² + 3960 t₁ - 44 t₁²

    18350 = 3960 t₁ - 22 t₁²

    9175 = 1980 t₁ - 11 t₁²

    11 t₁² - 1980 t₁ + 9175 = 0

    Using quadratic formula:

    t₁ = [ 1980 ± √ (1980² - 4 (11) (9175)) ] / 22

    t₁ = 4.76, 175

    Since t₁ can't be greater than 90, t₁ = 4.76 s.

    Therefore, t₂ = 85.2 s.

    And v = 44 t₁ = 209 ft/s.
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