(a) To determine To show: When the initial velocity vector bisects angle AOR , the downhill range will be the greatest. Explanation Sketch the given situation as shown in below Figure 1. When the projectile hits OR at time t , then tan β = y x . The distance downrange will be x = ( υ 0 cos ( α − β ) ) t and the height of the projectile is y = ( υ 0 sin ( α − β ) ) t − 1 2 g t 2 . Since R is below the ground level, y = ( υ 0 sin ( α − β ) ) t − 1 2 g t 2 < 0 . Let the absolute value | y | = 1 2 g t 2 − ( υ 0 sin ( α − β ) ) t > 0 . Then, tan β = 1 2 g t 2 − ( υ 0 sin ( α − β ) ) t ( υ 0 cos ( α − β ) ) t = 1 2 g t − ( υ 0 sin ( α − β ) ) ( υ 0 cos ( α − β ) ) Now obtain the time when the projectile hits the downhill slope as follows. ( υ 0 cos ( α − β ) ) tan β = 1 2 g t − ( υ 0 sin ( α − β ) ) 1 2 g t = ( υ 0 cos ( α − β ) ) tan β + ( υ 0 sin ( α − β ) ) t = 2 ( υ 0 cos ( α − β ) ) tan β + 2 ( υ 0 sin ( α − β ) ) g Substitute the obtained value of t in x = ( υ 0 cos ( α − β ) ) t as shown below (b) To determine The launch angle that maximizes the range if the projectile were fired uphill.

Thomas' Calculus: Early Transcendentals (14th Edition)

14th Edition

ISBN: 9780134439020

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 13.2, Problem 35E

(a)

To determine

To show: When the initial velocity vector bisects angle AOR, the downhill range will be the greatest.

(b)

To determine

The launch angle that maximizes the range if the projectile were fired uphill.

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Chapter 13.2, Problem 34E

Chapter 13.2, Problem 36E

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Chapter 13 Solutions

Thomas' Calculus: Early Transcendentals (14th Edition)

Ch. 13.1 - In Exercises 1–4, find the given limits. 1. Ch. 13.1 - In Exercises 1–4, find the given limits. 2. Ch. 13.1 - In Exercises 1–4, find the given limits. 3. Ch. 13.1 - In Exercises 1–4, find the given limits. 4. Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

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