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: