Consider a projectile subject to a drag F = -m a v. If it is fired with speed v

Question

Consider a projectile subject to a drag F = -m a v. If it is fired with speed v at an angle ?, it can be shown that the height as a function of time is given by (just accept this here) y(t) = 1/a(vsin? + (g/a))(1-e^(-at))-(gt/a) Show that this reduces to the usual projectile expression, y(t) = (vsin?)t-(g(t^2)/2), in the limit of small a. What exactly is meant by "small a"? What are the physical dimensions of the parameter a? What combination of parameters in this problem has the same units? Consider a projectile subject to a drag F = -m a v. If it is fired with speed v at an angle ?, it can be shown that the height as a function of time is given by (just accept this here) y(t) = 1/a(vsin? + (g/a))(1-e^(-at))-(gt/a) Show that this reduces to the usual projectile expression, y(t) = (vsin?)t-(g(t^2)/2), in the limit of small a. What exactly is meant by "small a"? What are the physical dimensions of the parameter a? What combination of parameters in this problem has the same units?

Explanation / Answer

Part 1) a is considered the drag coefficient, it is a number that varies due to the type of surface. There should be a list of these in your book, or online. But, a small drag coefficient is in the range of .04(streamlined body). So assuming that the drag coefficient is close to zero, you can eliminate its effect on the equation. So to show solving of equation, just plug in 0 for your "a". Part 2) To show that the Force (drag)=-mav, means that each factor in the equation has its own units. So if you set (Newtons)=(kilograms)x(?)x(meter/second), then you can solve for "?" and will come up with a baseline unit of measurement of "a".

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