(a) Explanation Given: An object of mass m is projected upward from ground level with initial velocity v 0 against air resistance proportional to its velocity v ( t ) . Thus, m d v d t = − m g − k v . Calculation: The objective is to solve the differential equation involving mass, velocity, height and air resistance of an object thrown upwards. And the differential equation is defined as m d v d t = − m g − k v …… (1) Where, m be the mass of the object, k be the air resistance, v 0 be the initial velocity, v be the velocity and g be the acceleration due to gravity. Now, as it is a first order ordinary differential equation so it can be solved by method of linear equations, Rewrite the equation (1) as d v d t + k v m = − g …… (2) Integrating factor is e ∫ k m d t . Thus, I F = e k m t . Multiply (2) on both sides by e k m t and get e k m t ⋅ d v d t + e k m t ⋅ k v m = − e k m t ⋅ g d ( v ⋅ e k m t ) d t = − e k m t ⋅ g d ( v ⋅ e k m t ) = − e k m t ⋅ g ⋅ d t Integrate both sides and get v ⋅ e k m t = − ∫ e k m t (b) To determine To find: when does the object reach its maximum height and explain the maximum height (c) To determine To find: the maximum height and the nearest second the object hit the ground (d) To determine To find: whether the object hit the ground sooner or later than the object in part c

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Differential Equations

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Chapter 7.6, Problem 59PS

(a)

To determine

To solve: the equation for v(t) , and then find s(t) the height of the object above the ground at time t .

(b)

To determine

To find: when does the object reach its maximum height and explain the maximum height

(c)

To determine

To find: the maximum height and the nearest second the object hit the ground

(d)

To determine

To find: whether the object hit the ground sooner or later than the object in part c

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Chapter 7.6, Problem 58PS

Chapter 7.6, Problem 60PS

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Chapter 7 Solutions

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the equation for v ( t ) , and then find s ( t ) the height of the object above the ground at time t .

Solution Summary: The objective is to solve the differential equation involving mass, velocity, height and air resistance of an object thrown upwards.

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To solve: the equation for v(t) , and then find s(t) the height of the object above the ground at time t .

To find: when does the object reach its maximum height and explain the maximum height

To find: the maximum height and the nearest second the object hit the ground

To find: whether the object hit the ground sooner or later than the object in part c

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Chapter 7 Solutions