Question

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

Answer 1

Answer:

A) $\frac{g}{b}(1-e^{-bt})$

Explanation:

Since a = g - bv,

We can substitute a = dv/dt into the equation.

Then, the equation will be like dv/dt = g - bv.

So we got first order differential equation.

As known, v = 0 at t = 0, and v = g/b at t = ∞.

Since $\frac{dv}{dt}= g - bv = b( \frac{g}{b} - v)$ ⇒ $\frac{dv}{ \frac{g}{b} - v}= bdt$

So take the integral of both side.

$- ln (\frac{g}{b} - v) = bt + C$

Since for t=0, v = 0 ⇒ $C =- ln (\frac{g}{b})$

$v = \frac{g}{b} + e^{-bt-ln(\frac{g}{b})} = \frac{g}{b}- \frac{g}{b}e^{-bt} = \frac{g}{b}(1-e^{-bt})$