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.