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/s^{2} 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 downdirection. Therefore, the acceleration, velocity, and displacement will all be positive in the downdirection. Also, for the solution, let d = 0.9g


Given: a = d  cv (m/s^{2}) 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 
usubstitution




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:



usubstitution




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.
