Question

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

Short answer

#tag_title#Step 2: Determine the vertical component of velocity#tag_content#Using the given initial velocity and angle in radians, we can find the vertical component of velocity (v_y) using the sine function: $$ v_y = v_0 \sin(\theta) $$ Where: $v_y$: vertical component of velocity $v_0$: initial velocity (40 m/s) $\theta$: angle in radians Plug in the values: $$ v_y = 40 \sin(56.7 \times \frac{\pi}{180}) $$ #tag_title#Step 3: Calculate the time to reach the highest point#tag_content#The time it takes for the football to reach its highest point can be calculated using the following equation: $$ t_{\text{up}} = \frac{v_y}{g} $$ Where: $t_{\text{up}}$: time to reach the highest point $v_y$: vertical component of velocity $g$: acceleration due to gravity (9.81 m/s²) Plug in the values: $$ t_{\text{up}} = \frac{v_y}{9.81} $$ #tag_title#Step 4: Calculate the total hang time#tag_content#The total hang time is twice the time it takes to reach the highest point since the time it takes to fall back to the ground will be the same as the time taken to reach the highest point: $$ t_{\text{hang}} = 2t_{\text{up}} $$ Plug in the value of $t_{\text{up}}$: $$ t_{\text{hang}} = 2(\frac{v_y}{9.81}) $$ Solve for $t_{\text{hang}}$ to find the total hang time of the punted football.

Step-by-step solution

Convert the angle to radians

It's easier to work with angles in radians when dealing with physics problems. Convert the angle from degrees to radians using the formula: radians = (degrees × π) / 180. The given angle is \(56.7^{\circ}\). We can convert it to radians as follows: $$ \text{radians }= (\text{angle in degrees})\times \frac{\pi}{180} $$ $$ \text{radians }= (56.7)\times \frac{\pi}{180} $$

Key concepts

Trajectory Analysis

Understanding the trajectory of an object in motion is crucial in physics. Trajectory refers to the path that an object follows through space as a function of time. When a football is punted, its trajectory is shaped by its initial velocity and the angle at which it is launched. These factors work together to define the arc the football travels from the kicker's foot to the ground.
A common way to analyze this trajectory in physics is by breaking it down into two components: horizontal and vertical motion.

• The horizontal motion is constant since the only force acting on it (assuming no air resistance) is gravity, which doesn’t affect horizontal speed.
• The vertical motion is affected by gravity, which pulls the football down, creating a symmetrical path.

By understanding both components, we can determine various factors like the maximum height, range, and, as the original exercise focuses on, the hang time — how long the football remains airborne before it lands.

Trigonometric Conversion

To solve physics problems involving angles, converting between degrees and radians is often essential. Degrees and radians are two units of measuring angles, with radians being more natural for many mathematical calculations. Therefore, problems like the football trajectory often require such conversion.
The conversion is simple: multiply the angle in degrees by \(\frac{\pi}{180}\) to get the angle in radians. For instance, in the original exercise, we had an angle of \(56.7^{\circ}\). The conversion to radians is calculated as follows:

• Use the formula: \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\).
• Plug in \(56.7\) for degrees: \(\text{radians} = 56.7 \times \frac{\pi}{180}\).

This conversion allows one to use the angle in trigonometric functions, which are required to analyze components of the trajectory in physics problems.

Physics Problem Solving

Physics problems, like calculating the hang time of a football, often follow a methodical approach. Solving these involves interpreting the problem statement carefully, identifying known and unknown quantities, and applying relevant formulas. Here's a step-by-step approach to problem-solving:

• Identify what you know: Initial velocity, angle of launch, and that gravity is acting downward.
• Break down the motion: Use physics concepts like projectile motion to split it into horizontal and vertical components.
• Apply the right formulas: Equations of motion, such as \(y = v_{0y} \cdot t + \frac{1}{2} a \cdot t^2\), where \(v_{0y}\) is the initial vertical velocity, help find the hang time.

Combining these steps with trigonometric conversions and careful calculation can solve the exercise, determining how long before the punted football hits the ground. Approach each problem calmly, ensuring each step logically follows from the last.