Suppose that a falling hailstone with density 8 = 1 starts from rest with negligible radius r = 0. Thereafter its ra- dius is r = kt (k is a constant) as it grows by accretion during its fall. Use Newton's second law-according to which the net force F acting on a possibly variable mass m equals the time rate of change dp/dt of its momentum p = mv-to set up and solve the initial value problem (mv) = mg, v(0) = 0, where m is the variable mass of the hailstone, v = dy/dt is its velocity, and the positive y-axis points downward. Then show that dv/dt = g/4. Thus the hailstone falls as though it were under one-fourth the influence of gravity.
Suppose that a falling hailstone with density 8 = 1 starts from rest with negligible radius r = 0. Thereafter its ra- dius is r = kt (k is a constant) as it grows by accretion during its fall. Use Newton's second law-according to which the net force F acting on a possibly variable mass m equals the time rate of change dp/dt of its momentum p = mv-to set up and solve the initial value problem (mv) = mg, v(0) = 0, where m is the variable mass of the hailstone, v = dy/dt is its velocity, and the positive y-axis points downward. Then show that dv/dt = g/4. Thus the hailstone falls as though it were under one-fourth the influence of gravity.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Suppose that a falling hailstone with density 8 = 1 starts
from rest with negligible radius r = 0. Thereafter its ra-
dius is r = kt (k is a constant) as it grows by accretion
during its fall. Use Newton's second law-according to
which the net force F acting on a possibly variable mass
m equals the time rate of change dp/dt of its momentum
p = mv-to set up and solve the initial value problem
(mv) = mg, v(0) = 0,
where m is the variable mass of the hailstone, v = dy/dt
is its velocity, and the positive y-axis points downward.
Then show that dv/dt = g/4. Thus the hailstone falls as
though it were under one-fourth the influence of gravity.
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