Question

(b) An asteroid of mass M falls through the Earth's atmosphere and reaches terminal velocity. Assume that the atmosphere is a turbulent fluid, so that a drag force is exerted on an object moving at velocity v. The quantity b is a constant. You can assume that the motion is sufficiently close to the Earth that the acceleration due to gravity is constant. (i) Explain what you understand by "turbulent flow" 121 (ii) Use Newton's Second Law to determine an expression for the terminal velocity 141 (iii) Does the drag force, or gravity do more work on the asteroid during its descent? Explain your answer. 141

Answer 1

(i) Turbulent flow in a fluid can be considered as a condition of flow in which parameters such as Velocity and pressure show random fluctuations with space and time.

(ii) Gravitational force on asteroid = Mg ( It is given that g can be assumed to be constant near the earth) ------ (1)

The drag force on asteroid = bv^2 ----------- (2)

The equation (1) tells us that as asteroid falls, it accelerates and thus v increases and consequently we can see from (2) that drag forces keep on increasing till such time ,the drag force exactly matches the gravitational force.

Mg = $bv_{t}^2$ , where $v_{t}$ is calle the terminal velocity since beyond this limit velocity can no longer increase ,since there is no net net external force

$v_{t}=\sqrt{Mg/b}$

(iii)

The gravity does net work on the asteroid till terminal velocity is reached- thereafter there is no net work., since at this stage the gravitational potential energy loss transforms into frictional loss in atmosphere.