(a) Explanation Given: m d v d t = m g − k v 2 . 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 2 …… (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 separable variable, Rewrite the equation (1) as d v d t = g − k v 2 m d v m g − k v 2 = d t m Integrate both sides and get ∫ d v m g − k v 2 = ∫ d t m 1 m g ∫ d v 1 − k m g v 2 = ∫ d t m 1 m g ∫ d v ( 1 − k m g v ) ( 1 + k m g v ) = ∫ d t m Now, use partial fractions on left hand side and get ∫ d t m = 1 m g ∫ d v ( 1 − k m g v ) ( 1 + k m g v ) ∫ d t m = 1 m g [ 1 2 ∫ d v ( 1 − k m g v ) + 1 2 ∫ d v ( 1 + k m g v ) ] ∫ d t m = 1 m g ⋅ 1 2 [ ( − m g k ) ln ( 1 − k m g v ) + ( m g k ) ln ( 1 + k m g v ) ] ∫ d t m = 1 m g ⋅ 1 2 ⋅ m g k [ − (b) To determine To find: the time taken by the object to hit the ground

7th Edition

ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Question

Chapter 7.6, Problem 60PS

(a)

To determine

To find: v(t) by solving the given differential equation and find the height s(t)

(b)

To determine

To find: the time taken by the object to hit the ground

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Chapter 7.6, Problem 59PS

Chapter 7.7, Problem 1PS

Chapter 7 Solutions

Calculus

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v ( t ) by solving the given differential equation and find the height s ( t )

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