Q. 33 Show that a ball thrown with an ... [FREE SOLUTION]

Chapter 11: Q. 33 (page 899)

Show that a ball thrown with an angle of elevation of $45^{鈭�}$ will travel farther than if it is thrown at any other angle.

Short Answer

Expert verified

The function $r (t) = <x (t), y (t)> r (t) = <(||v_{0}|| \cos 胃) t, - \frac{1}{2} g t^{2} + (||v_{0}|| \sin 胃) t>$ models the motion of the ball. The ball will hit the ground when the y-component is zero. This will occur when $t = \frac{2 ||v_{0}|| \sin 胃}{g}$ . At this time it will have travelled $\frac{||v_{0}|| \sin 胃 \cos 胃}{g}$ units horizontally. Thus, x-component will be maximized when $胃 = 45^{鈭�}$ .

Step by step solution

Step 1. Given information,

Consider the given question,

Angle of elevation is $45^{鈭�}$ .

Step 2. Find the distance traveled by the ball.

Assume the required function $r (t) = <x (t), y (t)>$ .

Assume $r (t) = <(||v_{0}|| \cos 胃) t, - \frac{1}{2} g t^{2} + (||v_{0}|| \sin 胃) t>$ models the motion of the ball.

If the ball hit the ground, y-component is zero.

$- \frac{1}{2} g t^{2} + (||v_{0}|| \sin 胃) t = 0 ||v_{0}|| \sin 胃 = \frac{1}{2} g t t = \frac{2 ||v_{0}|| \sin 胃}{g}$

The ball will travellocalid="1649613493436" $\frac{||v_{0}|| \sin 胃 \cos 胃}{g}$ units horizontally.

Step 3. Find when the x-component is maximum.

As $||v_{0}||, g$ are fixed, we fixed the value of $胃$ at which the x-component is maximum.

Assume $f (胃) = \sin 胃 \cos 胃$ . Then,

role="math" localid="1649613343908" $f (胃) = \frac{1}{2} (2 \sin 胃 \cos 胃) f (胃) = \frac{1}{2} \sin 2 胃 f' (0) = \frac{1}{2} (2 \cos 2 胃) f' (0) = \cos 2 胃$

Let $f' (胃) = 0$ . Then,

$\cos 2 胃 = 0 2 胃 = 90^{鈭�} 胃 = 45^{鈭�}$

Therefore, $\frac{||v_{0}|| \sin 胃 \cos 胃}{9}$ will be maximum when $胃 = 45^{鈭�}$ .

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91影视

Show that a ball thrown with an angle of elevation of $45^{鈭�}$ will travel farther than if it is thrown at any other angle.

Short Answer

Step by step solution

Step 1. Given information,

Step 2. Find the distance traveled by the ball.

Step 3. Find when the x-component is maximum.

One App. One Place for Learning.

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91影视

Show that a ball thrown with an angle of elevation of 45鈭� will travel farther than if it is thrown at any other angle.

Short Answer

Step by step solution

Step 1. Given information,

Step 2. Find the distance traveled by the ball.

Step 3. Find when the x-component is maximum.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Geometry

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Show that a ball thrown with an angle of elevation of $45^{鈭�}$ will travel farther than if it is thrown at any other angle.