Question

The force of resistance on an object moving through a fluid is proportional to the
square of its velocity. The initial velocity of an object of mass 1 kg moving through the
fluid is 40 ms−1
. After 5 s the velocity reduces to 20 ms−1
. Calculate its velocity after
10 s.

Answer 1

Write Newton's second law in differential form:

$$m\frac{dv}{dt}=-kv^2,$$

integrate:

$$-\frac{1}{v}=-\frac{k}{m}t+C.$$

Substituting v=40 m/s and t=0 find $C$:

$$C=-\frac{1}{40}\text{ s/m}.$$

Substituting v=20 m/s and t=5 s find $k$:

$$k=0.005\text{ kg/m}.$$

Substituting t=10 s determine what is required:

$$v=13.3\text{ m/s}.$$