62. analyze bernardo steps off a 3.0-m-high diving board and drops to the water below. at the same time michi jumps upward with a speed of 4.2 m/s from a 1.0-m-high diving board

General
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  1. analyze bernardo steps off a 3.0-m-high diving board and drops to the water below. at the same time michi jumps upward with a speed of 4.2 m/s from a 1.0-m-high diving board. taking the origin to be at the water’s surface and upward to be the positive x direction, write position-time equations for bernardo and michi.
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62. Analyze Bernardo steps off a 3.0-m-high diving board and drops to the water below. At the same time, Michi jumps upward with a speed of 4.2 m/s from a 1.0-m-high diving board. Taking the origin to be at the water’s surface and upward to be the positive x direction, write position-time equations for Bernardo and Michi.

Answer:
To solve this problem, we’ll use the equations of motion under constant acceleration due to gravity. Here is a step-by-step approach to derive the position-time equations for Bernardo and Michi.

  1. Bernardo’s Position-Time Equation:

    • Initial position, ( x_{B0} = 3.0 , \text{m} )
    • Initial velocity, ( v_{B0} = 0 , \text{m/s} ) (since he steps off the board and starts dropping)
    • Acceleration due to gravity, ( a = -9.8 , \text{m/s}^2 ) (negative because it’s directed downward)

    The position of Bernardo as a function of time ( t ) is given by the equation of motion:

    x_B(t) = x_{B0} + v_{B0} t + \frac{1}{2} a t^2

    Plugging in the values,

    x_B(t) = 3.0 \, \text{m} + 0 \cdot t + \frac{1}{2} (-9.8 \, \text{m/s}^2) t^2

    Simplifying,

    x_B(t) = 3.0 \, \text{m} - 4.9 \, \text{m/s}^2 \cdot t^2

  2. Michi’s Position-Time Equation:

    • Initial position, ( x_{M0} = 1.0 , \text{m} )
    • Initial velocity, ( v_{M0} = 4.2 , \text{m/s} ) (since she jumps upward)
    • Acceleration due to gravity, ( a = -9.8 , \text{m/s}^2 ) (negative because it’s directed downward)

    The position of Michi as a function of time ( t ) is given by the equation of motion:

    x_M(t) = x_{M0} + v_{M0} t + \frac{1}{2} a t^2

    Plugging in the values,

    x_M(t) = 1.0 \, \text{m} + 4.2 \, \text{m/s} \cdot t + \frac{1}{2} (-9.8 \, \text{m/s}^2) t^2

    Simplifying,

    x_M(t) = 1.0 \, \text{m} + 4.2 \, \text{m/s} \cdot t - 4.9 \, \text{m/s}^2 \cdot t^2

Final Answer:
The position-time equations for Bernardo and Michi are:

  • Bernardo:

    x_B(t) = 3.0 \, \text{m} - 4.9 \, \text{m/s}^2 \cdot t^2

  • Michi:

    x_M(t) = 1.0 \, \text{m} + 4.2 \, \text{m/s} \cdot t - 4.9 \, \text{m/s}^2 \cdot t^2

These equations describe their respective positions as functions of time after they start their respective motions.