Physics- Work Energy and Power

Classical Mechanics Level 2

A raindrop of mass m m is falling vertically through the air with a steady speed of v . v. The raindrop experiences a retarding force of k v kv due to the air, where k k is a constant. The acceleration of free fall is g . g.

Which expression gives the kinetic energy of the raindrop?

m 3 g 2 2 k 2 \frac{m^3g^2}{2k^2} m g k \frac{mg}{k} m g 2 2 k 2 \frac{mg^2}{2k^2} m 3 g 2 k 2 \frac{m^3g^2}{k^2}

Koay Xj by koay xj , 21, Malaysia

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1 solution

Tom Engelsman
Aug 18, 2018

Let us model the raindrop's free-fall motion via the following differential equation:

m ⋅ d 2 x d t 2 = m g − k v m \cdot \frac{d^2x}{dt^2} = mg - kv .

where x(t) is the raindrop's position with respect to time t. Since the raindrop falls at a steady speed, its acceleration is therefore zero (i.e. d 2 x d t 2 = 0 \frac{d^2x}{dt^2} = 0 ). Our differential equation reduces to 0 = m g − k v 0 = mg - kv , and solving for the speed gives v = m g k v = \frac{mg}{k} . The kinetic energy of the drop finally computes to:

K E = 1 2 m v 2 = 1 2 m ( m g k ) 2 = m 3 g 2 2 k 2 . KE = \frac{1}{2}mv^2 = \frac{1}{2}m(\frac{mg}{k})^2 = \boxed{\frac{m^3g^2}{2k^2}}.

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