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When the mass m of a body is changing with time, Newton’s

A First Course in Differential Equations with Modeling Applications | 10th Edition | ISBN: 9781111827052 | Authors: Dennis G. Zill ISBN: 9781111827052 44

Solution for problem 22E Chapter 1.3

A First Course in Differential Equations with Modeling Applications | 10th Edition

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A First Course in Differential Equations with Modeling Applications | 10th Edition | ISBN: 9781111827052 | Authors: Dennis G. Zill

A First Course in Differential Equations with Modeling Applications | 10th Edition

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Problem 22E

When the mass m of a body is changing with time, Newton’s second law of motion becomes

where F is the net force acting on the body and mv is its momentum. Use (17) in Problems 21 and 22.

In Problem 21, the mass m(t) is the sum of three different masses: where mp is the constant mass of the payload, mv is the constant mass of the vehicle, and mf (t) is the variable amount of fuel.

(a) Show that the rate at which the total mass m(t) of the rocket changes is the same as the rate at which the mass mf (t) of the fuel changes.

(b) If the rocket consumes its fuel at a constant rate , find m(t). Then rewrite the differential equation Problem 21 in terms of  and the initial total mass m(0) = m0.

(c) Under the assumption in part (b), show that the burnout time tb >0 of the rocket, or the time at which all the fuel is consumed, is  where mf (0) is the initial mass of the fuel.

(reference problem 21)

When the mass m of a body is changing with time, Newton’s second law of motion becomes

where F is the net force acting on the body and mv is it momentum. Use (17) in Problems 21 and 22.

A small single-stage rocket is launched vertically as shown in Figure 1.3.19. Once launched, the rocket consumes its fuel, and so its total mass m(t) varies with time t > 0. If it is assumed that the positive direction is upward, air resistance is proportional to the instantaneous velocity v of the rocket, and R is the upward thrust or force generated by the propulsion system, then construct a mathematical model for the velocity v(t) of the rocket. [Hint: See (14) in Section 1.3.]

(reference problem 14)

The right-circular conical tank shown in Figure 1.3.13 loses water out of a circular hole at its bottom. Determine a differential equation for the height of the water h at time t > 0. The radius of the hole is 2 in., g = 32 ft/s2, and the friction/contraction factor introduced in Problem 13 is c = 0.6.

(reference problem 13)

Suppose water is leaking from a tank through a circular hole of area Ah at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to is an empirical constant. Determine a differential equation for the height h of water at time t for the cubical tank shown in Figure 1.3.12. The radius of the hole is 2 in., and g =32 ft/s2.

Step-by-Step Solution:

Solution

Step 1

In this problem, we have to show that the total mass of the rocket changes is the same as the rate at which the mass of the fuel changes.

Step 2 of 4

Chapter 1.3, Problem 22E is Solved
Step 3 of 4

Textbook: A First Course in Differential Equations with Modeling Applications
Edition: 10
Author: Dennis G. Zill
ISBN: 9781111827052

Since the solution to 22E from 1.3 chapter was answered, more than 754 students have viewed the full step-by-step answer. This full solution covers the following key subjects: mass, rocket, hole, Water, Fuel. This expansive textbook survival guide covers 109 chapters, and 4053 solutions. This textbook survival guide was created for the textbook: A First Course in Differential Equations with Modeling Applications, edition: 10. The full step-by-step solution to problem: 22E from chapter: 1.3 was answered by , our top Calculus solution expert on 07/17/17, 09:41AM. A First Course in Differential Equations with Modeling Applications was written by and is associated to the ISBN: 9781111827052. The answer to “When the mass m of a body is changing with time, Newton’s second law of motion becomes where F is the net force acting on the body and mv is its momentum. Use (17) in 21 and 22.In 21, the mass m(t) is the sum of three different masses: where mp is the constant mass of the payload, mv is the constant mass of the vehicle, and mf (t) is the variable amount of fuel.(a) Show that the rate at which the total mass m(t) of the rocket changes is the same as the rate at which the mass mf (t) of the fuel changes.(b) If the rocket consumes its fuel at a constant rate , find m(t). Then rewrite the differential equation in terms of and the initial total mass m(0) = m0.(c) Under the assumption in part (b), show that the burnout time tb >0 of the rocket, or the time at which all the fuel is consumed, is where mf (0) is the initial mass of the fuel.(reference problem 21)When the mass m of a body is changing with time, Newton’s second law of motion becomes where F is the net force acting on the body and mv is it momentum. Use (17) in 21 and 22.A small single-stage rocket is launched vertically as shown in Figure 1.3.19. Once launched, the rocket consumes its fuel, and so its total mass m(t) varies with time t > 0. If it is assumed that the positive direction is upward, air resistance is proportional to the instantaneous velocity v of the rocket, and R is the upward thrust or force generated by the propulsion system, then construct a mathematical model for the velocity v(t) of the rocket. [Hint: See (14) in Section 1.3.] (reference problem 14)The right-circular conical tank shown in Figure 1.3.13 loses water out of a circular hole at its bottom. Determine a differential equation for the height of the water h at time t > 0. The radius of the hole is 2 in., g = 32 ft/s2, and the friction/contraction factor introduced in is c = 0.6. (reference problem 13)Suppose water is leaking from a tank through a circular hole of area Ah at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to is an empirical constant. Determine a differential equation for the height h of water at time t for the cubical tank shown in Figure 1.3.12. The radius of the hole is 2 in., and g =32 ft/s2.” is broken down into a number of easy to follow steps, and 433 words.

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