Explanation Given: A projectile is launched at an angle of θ hits an h-meters tower at a distance of d meters away. Proof: Let the launch point is at the origin, and r ( t ) be the position vector . Let the acceleration due to gravity be g. Then, v ( t ) = ∫ a ( t ) d t = 〈 0 , − g t 〉 + C 0 . ......... (1) Let the projectile was launched with speed of v 0 at an angle of θ . So, it gives v ( 0 ) = 〈 v 0 cos θ , v 0 sin θ 〉 . From (1), v ( 0 ) = 〈 0 , − g ⋅ 0 〉 + C 0 v ( 0 ) = 〈 0 , 0 〉 + C 0 v ( 0 ) = C 0 C 0 = 〈 v 0 cos θ , v 0 sin θ 〉 Therefore, the equation (1) becomes, v ( t ) = 〈 0 , − g t 〉 + 〈 v 0 cos θ , v 0 sin θ 〉 = 〈 v 0 cos θ , v 0 sin θ − g t 〉 Then, r ( t ) = ∫ 〈 v 0 cos θ , v 0 sin θ − g t 〉 r ( t ) = 〈 v 0 cos θ t , v 0 sin θ t − g 2 t 2 〉 + C 1 At t = 0 , r ( t ) = 〈 0 , 0 〉 . So, the equation r ( t ) = 〈 v 0 cos θ t , v 0 sin θ t − g 2 t 2 〉 + C 1 gives C 1 = 0

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

See similar textbooks

Videos

Question

Chapter 13.5, Problem 24E

To determine

To show that when a projectile launched at an angle θ will hit the top of an tower of height h-meter located at a distance of d meters away if the initial speed is v0=g/2dsecθdtanθ−h.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.5, Problem 23E

Chapter 13.5, Problem 25E

Students have asked these similar questions

Total marks 15 3. (i) Let FRN Rm be a mapping and x = RN is a given point. Which of the following statements are true? Construct counterex- amples for any that are false. (a) If F is continuous at x then F is differentiable at x. (b) If F is differentiable at x then F is continuous at x. If F is differentiable at x then F has all 1st order partial (c) derivatives at x. (d) If all 1st order partial derivatives of F exist and are con- tinuous on RN then F is differentiable at x. [5 Marks] (ii) Let mappings F= (F1, F2) R³ → R² and G=(G1, G2) R² → R² : be defined by F₁ (x1, x2, x3) = x1 + x², G1(1, 2) = 31, F2(x1, x2, x3) = x² + x3, G2(1, 2)=sin(1+ y2). By using the chain rule, calculate the Jacobian matrix of the mapping GoF R3 R², i.e., JGoF(x1, x2, x3). What is JGOF(0, 0, 0)? (iii) [7 Marks] Give reasons why the mapping Go F is differentiable at (0, 0, 0) R³ and determine the derivative matrix D(GF)(0, 0, 0). [3 Marks]

5. (i) Let f R2 R be defined by f(x1, x2) = x² - 4x1x2 + 2x3. Find all local minima of f on R². (ii) [10 Marks] Give an example of a function f: R2 R which is not bounded above and has exactly one critical point, which is a minimum. Justify briefly Total marks 15 your answer. [5 Marks]

Total marks 15 4. : Let f R2 R be defined by f(x1, x2) = 2x²- 8x1x2+4x+2. Find all local minima of f on R². [10 Marks] (ii) Give an example of a function f R2 R which is neither bounded below nor bounded above, and has no critical point. Justify briefly your answer. [5 Marks]

Chapter 13 Solutions

Calculus: Early Transcendentals

Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

To show that when a projectile launched at an angle θ will hit the top of an tower of height h-meter located at a distance of d meters away if the initial speed is v 0 = g / 2 d sec θ d tan θ − h .

Solution Summary: The author illustrates how a projectile launched at an angle of theta hits an h-meters tower at distance of d meters away.

Videos

To show that when a projectile launched at an angle θ will hit the top of an tower of height h-meter located at a distance of d meters away if the initial speed is v0=g/2dsecθdtanθ−h.

Want to see the full answer?

Chapter 13 Solutions