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Unformatted text preview: What Goes Up Must Come Down; Will Air Resistance Make It Return Sooner, or Later? John Lekner Victoria University Wellington, New Zeland Mathematics Magazine, January, 1982, Volume 55, Number 1, pp. 2628. A ball thrown straight up with speed would, in the absence of air, return in time Air resistance, or drag, will influence the return time in two ways: the maximum height reached is less than the zero-drag height and the speed at any height is less than the zero-drag speed. (These statements follow from the energy equation where is the mass of the ball, and is the (positive) work done against air resistance. The speed is zero at the top of the trajectory, so and at any Note that the energy conservation equation is not an additional physical principle: it follows from the equation of motion on multiplying by and integrating.) Thus with air resistance, the ball has a shorter distance to travel, but at a slower speed. What effect wins? Let be the deceleration due to the drag force. The equation of motion then reads on the way up, and on the way down (it is convenient to deal with speeds rather than velocities in this context). We will assume that has the property that there is just one speed at which the gravitational and drag forces are in balance. This defines the terminal speed The terminal speed is a natural scaling parameter for this problem. Let and Then by integrating (obtained from the equation of motion) we find the time to go up to maximum height is...
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