Question

A football player kicks a ball with a speed of \(22.4 \mathrm{~m} / \mathrm{s}\) at an angle of \(49^{\circ}\) above the horizontal from a distance of \(39 \mathrm{~m}\) from the goal line. a) By how much does the ball clear or fall short of clearing the crossbar of the goalpost if that bar is \(3.05 \mathrm{~m}\) high? b) What is the vertical velocity of the ball at the time it reaches the goalpost?

Short answer

Question: Calculate the difference in height between the ball's path and the crossbar and the vertical velocity of the ball when it reaches the goalpost, given an initial velocity of 22.4 m/s at an angle of 49° and a goal line distance of 39 m. Answer: 1. Find the horizontal and vertical components of the initial velocity: \(V_{0x}=V_{0} \cos{49°}\) \(V_{0y}=V_{0} \sin{49°}\) 2. Calculate the time it takes to reach the goal line: \(t = \frac{39\mathrm{~m}}{V_{0x}}\) 3. Find the vertical position of the ball when it reaches the goal line: \(h = V_{0y}t - \frac{1}{2}gt^2\) 4. Calculate the difference in height between the ball's path and the crossbar: \(\Delta h = h - 3.05\mathrm{~m}\) 5. Calculate the vertical velocity when the ball reaches the goalpost: \(v_y = V_{0y} - gt\) Use these steps to determine \(\Delta h\) and \(v_y\).

Step-by-step solution

1. Find the horizontal and vertical components of the initial velocity

Using the given initial velocity, \(22.4 \mathrm{~m} / \mathrm{s}\), and the angle, \(49^{\circ}\), find the horizontal and vertical components of the initial velocity using the following equations: \(V_{0x}=V_{0} \cos{49°}\) \(V_{0y}=V_{0} \sin{49°}\)

2. Calculate the time it takes to reach the goal line

Use the horizontal component of the initial velocity, \(V_{0x}\), to find the time it takes for the ball to reach the goal line. Use the equation: \(t = \frac{39\mathrm{~m}}{V_{0x}}\)

3. Find the vertical position of the ball when it reaches the goal line

Use the following kinematic equation to find the vertical position, \(h\), of the ball at the goal line: \(h = V_{0y}t - \frac{1}{2}gt^2\)

4. Calculate the difference in height between the ball's path and the crossbar

Subtract the height of the crossbar, \(3.05\mathrm{~m}\), from the calculated height \(h\) to find the difference: \(\Delta h = h - 3.05\mathrm{~m}\)

5. Calculate the vertical velocity when the ball reaches the goalpost

Using the vertical component of the initial velocity, \(V_{0y}\), and the time it takes to reach the goalpost, determine the ball's vertical velocity, \(v_y\), using the equation: \(v_y = V_{0y} - gt\) Now you have the difference in height between the ball's path and the crossbar and the vertical velocity of the ball when it reaches the goalpost.

Key concepts

Kinematic Equations

Understanding kinematic equations is essential for solving a wide range of physics problems, especially those related to projectile motion. These equations describe the motion of an object under the influence of gravity, and they are used to calculate positions, velocities, and time without considering the forces that cause such motion.

For projectile motion, the two main equations used are:

• The equation for the vertical position over time: \( h = V_{0y}t - \frac{1}{2}gt^2 \)
• The equation for the vertical velocity over time: \( v_y = V_{0y} - gt \)

Here, \( h \) is the height, \( V_{0y} \) is the initial vertical velocity, \( t \) is the time, \( g \) is the acceleration due to gravity (approximately \(9.81 m/s^2\) on Earth), and \( v_y \) is the vertical velocity at time \( t \). By mastering these equations, you can tackle any problem involving objects moving through the air, such as the case of the kicked football in the exercise.

Initial Velocity Components

The initial velocity of the projectile is crucial for trajectory calculations. Since the motion is two-dimensional, the initial velocity must be split into horizontal and vertical components. This step is necessary because gravity only affects the vertical motion, leaving the horizontal motion at a constant velocity.

The horizontal and vertical components of the initial velocity are found using trigonometry:

• Horizontal component: \( V_{0x} = V_0 \cos(\theta) \)
• Vertical component: \( V_{0y} = V_0 \sin(\theta) \)

Where \( V_0 \) is the initial velocity and \( \theta \) is the angle of launch. In the exercise, the ball is kicked at a \(49^\circ\) angle, meaning that we use sinus and cosines of \(49^\circ\) to determine the initial velocity components necessary for further calculations.

Trajectory Calculation

Calculating the trajectory of a projectile involves determining the path it will follow from launch until it lands or reaches a particular point, such as the goalpost in our example. The trajectory is influenced by both the initial velocity components and gravity.

To calculate the trajectory, you'll often need to determine the time of flight, the maximum height, and the horizontal distance traveled. Key steps include:

• Calculating the time it takes for the projectile to reach a specific point horizontally using the horizontal velocity and distance.
• Finding the vertical position at that time using the initial vertical velocity, time of flight, and the effect of gravity.

In the football problem, the horizontal distance to the goalpost and the initial horizontal velocity are used to compute the time of flight. Subsequently, this time is plugged into the second kinematic equation to find out how high the ball is above the crossbar when it reaches the goal line.

Physics Problem Solving

Problem solving in physics, particularly for projectile motion, involves a systematic approach that incorporates identifying the known variables, selecting the appropriate physical principles (such as kinematic equations), and executing computational steps. Here's a basic approach for successful problem-solving:

• Read and understand the problem, visualizing the situation.
• Identify and list all given quantities alongside what you're solving for.
• Choose the relevant physics principles and equations that apply to the problem.
• Carry out the calculations in a step-by-step manner, checking units for consistency and reasonability of results.

For instance, in our projectile motion problem, careful reading and understanding the situation allowed us to decompose the initial velocity into its components, apply kinematic equations, and ultimately establish whether the football clears the goalpost crossbar and with what vertical velocity it does so at that point.