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A raindrop of mass m and speed v collects water as it falls through a cloud. Show that, neglecting air resistance, mg = mv+ mv . Assume that the drop remains spherical and that the rate of accretion is proportional to the cross-sectional area of the drop multiplied by the speed of fall. Show that this implies that m = cvn3 %3D for some constant c. Using this equation, writing dm/dx = (dm/dt)/(dx/dt), and assuming that the drop starts from rest when it is infinitesimally small, find m as a function of x. Hence show that (:^)P - 2g = 0 . Finally, multiply through by a particular power of x (i.e. x" for some value of n) so that the first two terms combine to form a perfect derivative, and show that the acceleration of the raindrop is constant and equal to g/7. + 6 dr

A raindrop of mass m and speed v collects water as it falls through a cloud. Show that, neglecting air resistance, mg = mv+ mv . Assume that the drop remains spherical and that the rate of accretion is proportional to the cross-sectional area of the drop multiplied by the speed of fall. Show that this implies that m = cvn3 %3D for some constant c. Using this equation, writing dm/dx = (dm/dt)/(dx/dt), and assuming that the drop starts from rest when it is infinitesimally small, find m as a function of x. Hence show that (:^)P - 2g = 0 . Finally, multiply through by a particular power of x (i.e. x" for some value of n) so that the first two terms combine to form a perfect derivative, and show that the acceleration of the raindrop is constant and equal to g/7. + 6 dr

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A raindrop of mass m and speed v collects water as it falls through a cloud. Show that, neglecting air resistance, mg = mv + mv. Assume that the drop remains spherical and that the rate of accretion is proportional to the cross-sectional area of the drop multiplied by the speed of fall. Show that this implies that m = cvm3 , 2/3 for some constant c. Using this equation, writing dm/dx = (dm/dt)/(dx/dt), and assuming that the drop starts from rest when it is infinitesimally small, find m as a function of x. Hence show that d(v²). v² + 6 dr 2g = 0 . Finally, multiply through by a particular power of x (i.e. x" for some value of n) so that the first two terms combine to form a perfect derivative, and show that the acceleration of the raindrop is constant and equal to g/7.
Transcribed Image Text:A raindrop of mass m and speed v collects water as it falls through a cloud. Show that, neglecting air resistance, mg = mv + mv. Assume that the drop remains spherical and that the rate of accretion is proportional to the cross-sectional area of the drop multiplied by the speed of fall. Show that this implies that m = cvm3 , 2/3 for some constant c. Using this equation, writing dm/dx = (dm/dt)/(dx/dt), and assuming that the drop starts from rest when it is infinitesimally small, find m as a function of x. Hence show that d(v²). v² + 6 dr 2g = 0 . Finally, multiply through by a particular power of x (i.e. x" for some value of n) so that the first two terms combine to form a perfect derivative, and show that the acceleration of the raindrop is constant and equal to g/7.
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