Question

Ask you to solve a variety of problems based on the principles of projectile motion. A football quarterback throws a pass from a height of \(6 \mathrm{ft}\), intending to hit his receiver 20 yds away at a height of \(5 \mathrm{ft}\). (a) If the ball is thrown at a rate of \(50 \mathrm{mph}\), what angle of elevation is needed to hit his intended target? (b) If the ball is thrown at with an angle of elevation of \(8^{\circ},\) what initial ball speed is needed to hit his target?

Short answer

(a) The angle of elevation needed is approximately \( 19.37^{\circ} \). (b) The initial speed needed is approximately 90.25 ft/s.

Step-by-step solution

Convert Units and Define Variables

First, convert the distance to the target from yards to feet: 20 yards = 60 feet. The height from which the ball is thrown is 6 feet, and the target's height is 5 feet. Convert the initial velocity: 50 mph is approximately 73.33 feet/second. Define variables: initial speed \( v_0 = 73.33 \text{ ft/s} \), horizontal distance \( x = 60 \text{ ft} \), initial height \( h_0 = 6 \text{ ft} \), and final height \( h_f = 5 \text{ ft} \).

Use the Projectile Motion Equation

The equation for projectile motion is \( h = h_0 + x \tan{\theta} - \frac{g x^2}{2(v_0 \cos{\theta})^2} \), where \( h \) is the final height, \( g = 32.2 \text{ ft/s}^2 \) is the acceleration due to gravity. For part (a), set \( h = 5 \text{ ft} \) and solve for \( \theta \).

Calculate for Angle of Elevation (Part a)

Rearrange the equation: \( 5 = 6 + 60 \tan{\theta} - \frac{32.2 \times 60^2}{2(73.33 \cos{\theta})^2} \). Simplify and solve this equation using numerical methods (e.g., graphing calculator or computational tool) to find \( \theta \). The calculated angle of elevation will be approximately \( 19.37^{\circ} \).

Setup for Initial Speed Calculation (Part b)

For part (b), let \( \theta = 8^{\circ} \) and solve the same projectile motion equation for \( v_0 \) with \( x = 60 \) ft, \( h = 5 \) ft, and \( h_0 = 6 \) ft.

Calculate Initial Speed (Part b)

Using the equation \( 5 = 6 + 60 \tan{8^{\circ}} - \frac{32.2 \times 60^2}{2(v_0 \cos{8^{\circ}})^2} \), isolate \( v_0 \). Numerical methods will yield \( v_0 \approx 90.25 \text{ ft/s} \) (convert to mph if needed).

Key concepts

Angle of Elevation

The angle of elevation in projectile motion is crucial in determining the path of an object, such as a football in this case, when it is launched. Essentially, it refers to the angle at which the object is projected above the horizontal plane. This angle significantly affects the trajectory and the distance the projectile will cover.

In our example, determining the correct angle of elevation is essential for ensuring that the football reaches the receiver accurately. With the precise angle, one can predict how high and how far the object will go before it lands. In our exercise, the quarterback needs to throw the ball at an angle of approximately \( 19.37^{\circ} \) to reach the target.

Understanding the angle of elevation is important not only in sports but also in various fields requiring precise projectile launches. Proper calculation ensures that the thrown object, whether it be a football, arrow, or any projectile, hits the target accurately.

Initial Velocity

Initial velocity is another key component in projectile motion. It refers to the speed at which an object begins its journey. The initial speed helps determine how far and how fast a projectile will travel before reaching its destination.

In the scenario provided, the quarterback knows he needs to throw the ball with an initial velocity of 73.33 feet per second (or converted from mph) to get it to his intended receiver if the angle is correct. Should the angle change, as demonstrated in part b, the initial velocity must be adjusted accordingly to ensure accuracy.

Grasping the concept of initial velocity can help you predict the object's future behavior, making it an integral aspect when calculating projectile motion scenarios.

Trajectory Analysis

Trajectory is the path that a projectile follows, and analyzing it involves understanding all the forces acting on the object, including gravity and initial launch conditions. The trajectory in projectile motion is usually parabolic due to gravity's influence pulling the object down after it is launched.

For effective trajectory analysis, we use the projectile motion equation, which incorporates angle, initial speed, and gravity to predict the object's path. In this particular exercise, solving the motion equation allows us to forecast where the football will land.

• A steeper angle of elevation results in a higher, shorter arc.
• A shallower angle results in a longer, flatter arc.

Understanding trajectory is indispensable not only in sports but also in fields like engineering and physics, where predicting an object's path is crucial.

Physics in Sports

Physics plays a pivotal role in sports, providing explanations for the mechanics behind various athletic actions. Understanding physics principles such as projectile motion can enhance both the performance and enjoyment of sports like football, baseball, and basketball.

In our example, a quarterback throwing a ball implements principles of physics to optimize his throw. The angle of elevation, initial velocity, and gravity's impact are all tuned to ensure that the ball reaches the receiver accurately. This practical application of physics demonstrates how athletes intuitively or professionally utilize scientific concepts to improve their gameplay.

By studying projectile motion in sports, you gain insight into how athletes calculate speed, improve accuracy, and manage different environmental factors. This knowledge empowers players and coaches to optimize sports strategies and techniques efficiently.