Problem 13.46 Page 41 The greatest ocean depth yet discovered is the Marianas Trench in the western Pacific Ocean. A steel ball released at the surface requires 64 minutes to reach the bottom. The ball's downward acceleration is a=0.9g - cv, where g=9.81 m/s2 and the constant c= 3.02 s -1. What is the depth of the Marianas Trench in kilometers? For the solution, the surface of the ocean is considered 0 for the displacement. The acceleration will be positive in the down-direction. Therefore, the acceleration, velocity, and displacement will all be positive in the down-direction. Also, for the solution, let d = 0.9g Given: a = d - cv (m/s2) c = 3.02 s-1 t = 64 minutes = 3840 seconds Definition of acceleration Substitute accel. and multiply by dt Divide by accel. to separate variables To integrate the right side, use the rule let u = d - c*v du = -c dv u-substitution After integration At t = 0, v = 0 Initial conditions Evaluate the constant of integration Equation for time Isolate the natural log Use both sides as exponents for e Subtract d from both sides Divide by -c Definition of velocity Substitute v Multiply by dt To integrate the middle integral, use let: u-substitution After integration Initial conditions At t = 0, s = 0 Evaluate the constant of integration Equation of displacement At t = 3840 seconds The exponent of e evaluates to -11595. Since it is negative e will be in the denominator, and since the magnitude is so large this factor will be close enough to 0 that it can be ignored. Therefore, the Marianas Trench is 11.2 kilometers deep.