A raindrop falls through a cloud, increasing in size as it picks up moisture. Assume that its shape always remains spherical. Also assume that the rate of increase of its volume with respect to distance fallen is proportional to the cross-sectional area of the drop at any time (that is, the mass increase dm = ρdV is proportional to the volume πr2 dy swept out by the drop as it falls a distance dy). Show that the radius r of the drop is proportional to the distance y the drop has fallen if r = 0 when y = 0. Recall that when m is not constant, Newton’s second law is properly stated as (d/dt)(mv) = F. Use this equation to find the distance y which the drop falls in time t under the force of gravity, if y = ˙y = 0 at t = 0. Show that the acceleration of the drop is g/7 where g is the acceleration of gravity.
A raindrop falls through a cloud, increasing in size as it picks up moisture. Assume that its shape always remains spherical. Also assume that the rate of increase of its volume with respect to distance fallen is proportional to the cross-sectional area of the drop at any time (that is, the mass increase dm = ρdV is proportional to the volume πr2 dy swept out by the drop as it falls a distance dy). Show that the radius r of the drop is proportional to the distance y the drop has fallen if r = 0 when y = 0. Recall that when m is not constant, Newton’s second law is properly stated as (d/dt)(mv) = F. Use this equation to find the distance y which the drop falls in time t under the force of gravity, if y = ˙y = 0 at t = 0. Show that the acceleration of the drop is g/7 where g is the acceleration of gravity.
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