In a pollution control experiment, minute solid particles (typical mass 1×10-13 slug) are dropped in air. the terminal speed of the particles is measured to be 0.2 ft/s. the drag force of these particles is given by fd=kv, where v is the instantaneous particle speed. find the value of the constant k. find the time required to reach 99 percent of terminal speed.

1)

The particle experiences two forces during its fall:

- The force of gravity, , pointing downward, where m is the mass of the particle and g is the acceleration of gravity

- The drag force, , pointing upward, where k is a constant and v is the instantaneous speed of the particle

When the particle reaches its terminal speed, the acceleration becomes zero: this means that the net force is also zero, so the two forces are balanced. Therefore we can write

where is the terminal velocity. We have the following data:

is the mass

is the acceleration

is the terminal speed

Solving for k, we find

2) 28.6 ms

We now want to find the time at which the particle reaches 99 % of the terminal speed, so a speed of

The net force acting on the particle is

So the acceleration is

This differential equation can be rewritten as

And by integrating on both sides and performing the calculation, we can get the final expression for the velocity as

We want to find the time t at which v = 0.198 ft/s, so we now need to re-arrange the formula for t. This gives:

And substituting all the values, we find:

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