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.