Program Explanation Given information: The hailstone starts from rest with density δ = 1 and a negligible small radius r = 0 . After the radius grows by accretion, as r = k t and the net force F equals the time rate of change d p d t of momentum p = m v . The differential equation is, d d t ( m v ) = m g with v ( 0 ) = 0 Here, m is the variable mass of hailstone. Concept used: The linear first order equation is, d y d x + P ( x ) y = Q ( x ) The solution of above equation is, y ( x ) e ∫ P ( x ) d x = ∫ ( Q ( x ) e ∫ P ( x ) d x ) d x + C Here, e ∫ P ( x ) d x is the integrating factor and C is the constant of integration. Explanation: From the Newton’s second law the differential equation is, d d t ( m v ) = m g ....... (1) Here, g is with the influence of gravity. The radius of hailstone is, r ( t ) = k t The volume of spherical hailstone is, V = 4 3 π { r ( t ) } 3 Substitute k t for r ( t ) in above equation as, V = 4 3 π ( k t ) 3 = 4 3 π k 3 t 3 The density of the hailstone is, δ = m V Since, the density of hailstone is 1, substitute 1 for δ and 4 3 π k 3 t 3 for V in above equation as, 1 = m 4 3 π k 3 t 3 m = 4 3 π k 3 t 3 The time rate of change of momentum p = m v can be written as, d d t ( m v ) = v ⋅ d m d t + m d v d t Substitute 4 3 π k 3 t 3 for m in above equation as, d d t ( m v ) = v ⋅ d d t ( 4 3 π k 3 t 3 ) + m d v d t = v ⋅ 4 3 π k 3 ( 3 t 2 ) + m d v d t = 4 v π k 3 t 2 + m d v d t Substitute 4 v π k 3 t 2 + m d v d t for d d t

MyLab Math with Pearson eText -- 24-Month Standalone Access Card -- For Differential Equations and Boundary Value Problems: Computing and Modeling Tech Update

5th Edition

ISBN: 9780134872971

Author: Edwards, C., Penney, David, Calvis

Publisher: PEARSON

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Recurrence relation

A recurrence relation or recursive relation is an equation that represents a function in terms of the values of its smaller inputs. Every recurrence relation T(n) is a recursive function of integer n and consists of a base case and a recursive case.

Question

Chapter 1.5, Problem 42P

Program Plan Intro

To explain:The solution of differential equation ddt(mv)=mg and to show that the time rate of change of velocity of hailstone is dvdt=g4 .

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Chapter 1.5, Problem 41P

Chapter 1.5, Problem 43P

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Chapter 1 Solutions

MyLab Math with Pearson eText -- 24-Month Standalone Access Card -- For Differential Equations and Boundary Value Problems: Computing and Modeling Tech Update

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The solution of differential equation d d t ( m v ) = m g and to show that the time rate of change of velocity of hailstone is d v d t = g 4 .

Solution Summary: The author explains the solution of the differential equation dt(mv)=mg and the time rate of change of velocity of hailstone.

Concept explainers

To explain:The solution of differential equation ddt(mv)=mg and to show that the time rate of change of velocity of hailstone is dvdt=g4 .

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Chapter 1 Solutions