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28 March, 00:43

In the vertical jump, an Kobe Bryant starts from a crouch and jumps upward to reach as high as possible. Even the best athletes spend little more than 1.00 s in the air (their "hang time"). Treat Kobe as a particle and let ymax be his maximum height above the floor. Note: this isn't the entire story since Kobe can twist and curl up in the air, but then we can no longer treat him as a particle.

To explain why he seems to hang in the air, calculate the ratio of the time he is above ymax/2 moving up to the time it takes him to go from the floor to that height. You may ignore air resistance.

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  1. 28 March, 01:11
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    the athlete spends 2.4 times more time at the upper part of his way than in the lower one.

    Explanation:

    Let's find the velocity V1 of an athlete to reach half of the maximum height equation

    V1 = v20 - 2gh = v20 - 2g (ymax) / 2

    Here, Vo is the initial velocity of athlete, v1 is the velocity of athlete at half the maximum height, g is the acceleration due to gravity, h=ymax / 2 is half of the maximum height.

    We can fund the maximum height that athlete can reach from the law of conservation of energy

    KE = PE

    1/2M v20 = mg ymax

    ymax = v20 / 2g

    Then, substituting ymax into the first equation we get

    V21 = v20 - v20/2 = v20/2

    V1 = V0/∫2, we can find the time that the athlete needs to reach the maximum height (ymax) from the kinematic equation

    V = V0 - gt

    Here, V is the final velocity of an athlete at the maximum height; V0 is the initial velocity of an athlete

    Since, V=0ms-1, we get t=V0/g

    Similarly, we can find the time t1 that an athlete needs to reach maximum height from the Ymax/2:

    T1 = V1/g = V0/g∫2

    So, it is obvious that the time to reach Ymax from Ymax/2 is nothing more than the difference between t and t1:

    t-t1 = V0/g (1-1/∫2)

    finally, we can calculate the ratio of the time he is above Ymax/2 to the time it takes him to go from floor to that height.

    T1/t-t1 = V0/g∫2V0 * g∫2/∫2-1 = 2.4

    Answer; the athlete spends 2.4 times more time at the upper part of his way than in the lower one.
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