An object is released from rest at time t = 0 and falls through the air, which exerts a resistive force such that the acceleration a of the object is given by a = g bv, where v is the object's speed and b is a constant. if limiting cases for large and small values of t are considered, which of the following is a possible expression for the speed of the object as an explicit function of time? a) v = g(1-e^-bt)/b b) v = (ge^bt)/b c) v = (g+a)t/b

The correct option for the expression of speed as an explicit function of time is option A

A) v = g·(1 - )/b

The reason why option A is correct is given as follows;

Known:

The initial velocity of the object at time t = 0 is v = 0 (object at rest)

The function that represents the acceleration is a = g - b·v

Where;

v = The speed of the object at the given instant

b = A constant term

By considering the limiting case for time t, we have;

At very large values of t, the velocity will increase such that we have;

therefore,   and

The given equation can be rewritten as follows, to express the equation in terms the velocity;

Therefore;

The above differential equation gives;

Which gives;

and  

Therefore

At t = 0, v = 0, therefore;

Which gives;

∴ v = g·(1 - )/b

The correct option is option (A)

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