# Drag force on a sphere [closed]

A sphere of mass $M$ and radius $R$ moves with speed $V$ through a region of space that contains particles of mass $m$ that are at rest. There are $n$ of these particles per unit volume. Assume $m$ is very small compared to $M$, and assume that the particles do not interact with each other. What is the drag force on the sphere?

I found this problem in the section of conservation of energy in David morin classical mechanics book.

To address this question I firstly thought of the mechanism by which an ideal gas imposes a dragging Force proportional to velocity gradient owing to internal collisions between the body and the gas molecules. But in this case the main difference is that the spheres of mass m are at rest initially so that this approach simply wont do.

Then this idea came to my mind that the body,as it propagates through the medium creates spherical waves in that medium just like the pressure waves in ideal gas and if this is the case then the drag force must be proportional with the acoustic pressure at the region where the body lies.As the position of the body is not fixed,say,at position A it created a spherical wave and now the body has reached point B and at point B the drag force should be according to my logic equal to the acoustic pressure due to the wave which it created while it was it at point A. Now I am failing to mathematically model this idea to turn it into an answer to the question. How can i do this?

## closed as off-topic by sammy gerbil, Jon Custer, Yashas, heather, Qmechanic♦Aug 16 '17 at 6:56

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• I would add the fluid dynamics tag. If you are familiar with comp!ex numbers, this is related: en.wikipedia.org/wiki/Joukowsky_transform – user163104 Aug 13 '17 at 15:36
• Think in terms of energy conservation and where and how the energy flows. There is momentum exchange and alternately heat production. Drag occurs in the latter. – docscience Aug 13 '17 at 15:42

The simple calculation is that if an element dA of a surface is inclined at an angle $\delta$ to the flow, it will be impacted by particles in a stream of cross section $\sin{\delta}dA$, having momentum $\rho V \sin{\delta}dA$, where $\rho$ is the density of the flow and $V$ is its velocity. The component of this momentum normal to the surface that arrives per unit time is $\rho V^2\sin^2{\delta}$ per unit area and this can be thought of as the increase $P$ in fluid pressure. The drag force is the integral of this over the frontal area of the body. For a sphere of radius $R$ this is $$\int_0^{\pi/2}\rho V^2\cos^2{\theta}(2\pi R\sin{\theta}(\cos{\theta}d\theta)=(\rho V^2/2)(\pi R^2)$$ where $\theta$ is measured from the axis of symmetry so that $\delta=\pi/2-\theta$. If elastic collisions are assumed, as is probably the case for the Ops question, the answer is twice as big.