Problem

Unless otherwise stated, in the following problems we assume that the gravitational force is constant with g = 9.81 m/sec2 in the MKS system and g = 32 ft/sec2 in the U.S. Customary System.

If the object in Problem 1 has a mass of 500 kg instead of 5 kg, when will it strike the ground? [Hint: Here the exponential term is too large to ignore. Use Newton’s method to approximate the time t when the object strikes the ground (see Appendix B).]

Step 1:

In this problem, we have to find the time of the object to hit the ground.

Step 2:

The given values are

Mass : m= 500 kg

Gravitational force :g=9.81 m/s2

Proportionality constant b=50 Ns/m

The buoyancy force (while falling downwards) acts downwards the drags force upwards

Therefore m=mg-bv ……………..(1)

Step 3:

The general equation for the force Force= mass x acceleration

Where the acceleration is derivative of the velocity i.e a=

Therefore

mg-bv=m

dt=

Therefore we will find the time t by integrating the above equation

dt=m

Now we get

t=[ln(mg-bv)-ln(mg)]

ln

Taking exponential on both sides of the above equation we get

mg e=mg-bv………..(2)