Question

A football is punted with an initial speed of \(27.5 \mathrm{~m} / \mathrm{s}\) and a launch angle of \(56.7^{\circ} .\) What is its hang time (the time until it hits the ground again)?

Short answer

To calculate the hang time, follow these steps: 1. Calculate the initial vertical velocity: \(v_{y} = 27.5\,\text{m/s} \cdot \sin(56.7 \cdot \pi/180)\) 2. Calculate the time it takes to reach the maximum height: \(t_{top} = \dfrac{v_{y}}{g}\) 3. Calculate the total hang time: \(t_{hang} = 2t_{top}\) By following these steps, you'll be able to find the hang time of the football in seconds.

Step-by-step solution

Calculate the initial vertical velocity

To find the initial vertical velocity, we can use the following equation: \(v_{y} = v \sin(\theta)\) where \(v_y\) is the initial vertical velocity, \(v\) is the initial speed (27.5 m/s), and \(\theta\) is the launch angle (56.7 degrees). Plug in the given values and convert the angle to radians: \(v_{y} = 27.5\,\text{m/s} \cdot \sin(56.7 \cdot \pi/180)\)

Calculate the time it takes to reach maximum height

At maximum height, the vertical velocity will be zero. We can use the following equation to find the time it takes to reach the top: \(t_{top} = \dfrac{v_{y}}{g}\) where \(t_{top}\) is the time to reach maximum height, \(v_{y}\) is the initial vertical velocity, and \(g\) is the gravitational constant (9.81 m/s²). Plug in the values and solve for \(t_{top}\): \(t_{top} = \dfrac{27.5\,\text{m/s} \cdot \sin(56.7\cdot \pi/180)}{9.81\,\text{m/s}^2}\)

Calculate the total hang time

The hang time is twice the value of \(t_{top}\) as it takes the same time to ascend and descend: \(t_{hang} = 2t_{top}\) Plug in the value of \(t_{top}\) and solve for \(t_{hang}\): \(t_{hang} = 2 \cdot \dfrac{27.5\,\text{m/s} \cdot \sin(56.7\cdot \pi/180)}{9.81\,\text{m/s}^2}\) Once computed, this will give the hang time of the football.

Key concepts

Initial Vertical Velocity

Understanding the initial vertical velocity is crucial in analyzing projectile motion. This velocity component determines how high and how long an object will stay in the air. When an object is launched at an angle, like a football during a punt, its velocity is divided into horizontal and vertical components. The initial vertical velocity is calculated using the sine function from trigonometry, indicated in the equation:
\(v_{y} = v \sin(\theta)\)
where \(v_y\) is the initial vertical velocity, \(v\) is the initial speed, and \(\theta\) is the launch angle. For a given initial speed and angle, the higher the value of \(\sin(\theta)\), the greater the initial vertical velocity. This component is independent from horizontal movements and is solely responsible for the projectile's ascent and subsequent descent due to gravity.

Maximum Height in Projectile Motion

The maximum height a projectile reaches is the peak of its trajectory. At this point, the vertical velocity is momentarily zero before gravity pulls it back down. Trigonometry and physics work together to reveal the time it takes to reach this height through the formula:
\(t_{top} = \frac{v_{y}}{g}\)
where \(v_{y}\) is the initial vertical velocity and \(g\) is the gravitational acceleration. This time, \(t_{top}\), directly affects the maximum height achieved, illustrating how integral initial vertical velocity and gravity are to projectile motion.
The maximum height can be found using the vertical motion formulas derived from the basic equations of kinematics, considering the final vertical velocity at maximum height to be zero.

Gravitational Acceleration

Gravitational acceleration, often denoted as \(g\), is the rate at which objects accelerate towards the Earth due to gravity, approximately \(9.81 \text{ m/s}^2\). In projectile motion, gravity is the force that decelerates the projectile on its way up and accelerates it on its way down. It's crucial for determining the time to reach maximum height, as well as the overall flight time. The impact of gravitational acceleration is constant and does not depend on the mass of the object or the initial velocity, under the assumption that air resistance is negligible. Understanding how \(g\) influences projectile motion helps in predicting the path and duration of a projectile's journey.

Trigonometry in Physics

Trigonometry is an essential part of physics, especially when analyzing projectile motion. It allows us to decompose the initial velocity of an object into vertical and horizontal components. The use of sine (for vertical) and cosine (for horizontal) functions are a result of treating the initial velocity as the hypotenuse of a right triangle, with the other two sides representing the components. The equations\(v_{x} = v \cos(\theta)\) and \(v_{y} = v \sin(\theta)\)encompass this relationship. Through trigonometry, we're able to understand the behavior of projectiles and solve for variables such as hang time, range, and maximum height.