Problem 63 - Algebra and Trigonometry

Question

Range of a Projectile If a projectile is fired with velocity \(v_{0}\) at an angle \(\theta,\) then its range, the horizontal distance it travels (in feet), is modeled by the function $$ R(\theta)=\frac{v_{0}^{2} \sin 2 \theta}{32} $$ (See page \(614 .\) ) If \(v_{0}=2200 \mathrm{ft} / \mathrm{s}\) , what angle (in degrees) should be chosen for the projectile to hit a target on the ground 5000 \(\mathrm{ft}\) away?

Step-by-Step Solution

Answer

The angle should be approximately 0.95 degrees.

Step 1: Substitute Known Values into the Equation

Given that the projectile's initial velocity \(v_0\) is 2200 ft/s and the range \(R\) is 5000 ft, substitute these values into the range equation:\[5000 = \frac{2200^2 \sin 2\theta}{32}\]

Step 2: Simplify the Equation

To simplify the equation, multiply both sides by 32 to eliminate the denominator:\[ 5000 \times 32 = 2200^2 \sin 2\theta \]This becomes:\[ 160000 = 4840000 \sin 2\theta \]

Step 3: Solve for \(\sin 2\theta\)

Divide both sides of the equation by 4840000 to solve for \(\sin 2\theta\):\[\sin 2\theta = \frac{160000}{4840000}\]\[\sin 2\theta \approx 0.03305785\]

Step 4: Find \(2\theta\) Using the Inverse Sine

Using the inverse sine function, calculate \(2\theta\):\[2\theta = \sin^{-1}(0.03305785)\]This gives:\[ 2\theta \approx 1.895^ ext{o}\]

Step 5: Solve for \(\theta\)

Divide the result for \(2\theta\) by 2 to find \(\theta\):\[\theta = \frac{1.895}{2}\]\[\theta \approx 0.9475^ ext{o}\]

Step 6: Finalize the solution

The angle \(\theta\) should be rounded to an appropriate degree for practical application.Thus:\[\theta \approx 0.95^ ext{o}\]Angle should be as precise as possible for desired range.

Key Concepts

Projectile Motion

Projectile motion is a fascinating subject that involves the study of objects launched into the air. These objects follow a curved path under the influence of gravity alone. Knowing the motion of projectiles can help us predict where they'll land.
This is crucial in many fields, such as sports and engineering. In the given exercise, we work with a formula that estimates how far a projectile travels. This distance, called the range, depends on the launching speed and angle.
The range equation used is derived from the basic principles of kinematics, adjusted for motion in a gravitational field. By launching the projectile at various angles, different ranges can be achieved. Thus, understanding projectile motion isn't just about solving equations. It's also about visualizing how different forces act on an object in flight. This information allows engineers, athletes, and others to optimize launches for their specific needs.

Trigonometric Functions

Trigonometric functions are vital when analyzing projectile motion. These functions, such as sine and cosine, help us understand angles and distances. In our problem, the sine function plays a key role in calculating the range of a projectile.
The equation uses \ \( \sin 2\theta \ \) to manage changes in angle, which crucially affects projectile range. These functions relate angles to ratios of sides in right triangles. They allow us to solve for distances and angles efficiently.
When working with projectile motion, the right angle calculations ensure accurate predictions of motion. Understanding how \ \( \sin 2\theta \ \) affects the projectile's path is vital. It helps us determine the optimal angles to achieve the maximum range. Without trigonometry, precise predictions of projectile paths would be almost impossible.

Kinematics

Kinematics is the branch of physics that deals with motion. It describes the motion of objects without considering the forces that cause this motion. For projectiles, it helps describe how they move over time.
In our range formula, kinematics helps us relate velocity, angle, time, and distance. By knowing initial speed and angle, we can predict where the projectile will land. This is done without including the complexity of forces such as air resistance.
The formula used in the problem simplifies real-world motion into manageable computations. It assumes ideal conditions, such as no air resistance and constant gravity.
By understanding kinematics, we're equipped to predict and analyze motion in various contexts. This is essential in fields like robotics, game development, and even traffic modeling. Without kinematics, we wouldn't be able to connect an object's position to its speed and time.