Question

Position and Velocity of a Ball. If a stationary ball is released at a height \(h_{o}\) above the surface of the Earth with a vertical velocity \(v_{o}\), the position and velocity of the ball as a function of time will be given by the equations $$ \begin{gathered} h(t)=\frac{1}{2} g t^{2}+v_{0} t+h_{0} \\ v(t)=g t+v_{0} \end{gathered} $$ where \(g\) is the acceleration due to gravity \(\left(-9.81 \mathrm{~m} / \mathrm{s}^{2}\right), h\) is the height above the surface of the Earth (assuming no air friction), and \(v\) is the vertical component of velocity. Write a MATLAB program that prompts a user for the initial height of the ball in meters and velocity of the ball in meters per second, and plots the height and velocity as a function of time. Be sure to include proper labels in your plots.

Short answer

To write a MATLAB program that prompts a user for the initial height $h_0$ and velocity $v_0$ of a ball, and plots the height and velocity as a function of time, follow these steps: 1. Take user inputs for initial height $h_0$ and velocity $v_0$: ```Matlab h0 = input('Enter the initial height of the ball in meters: '); v0 = input('Enter the initial vertical velocity of the ball in meters per second: '); ``` 2. Calculate time duration until the ball reaches the ground using the quadratic formula and choose the positive root to represent the time: ```Matlab g = 9.81; times = roots([(-1/2)*g, v0, h0]); t_final = max(times); ``` 3. Create a time vector and calculate the position and velocity at different time intervals using the given equations: ```Matlab time = linspace(0, t_final, 1000); positions = (-1/2)*g*time.^2 + v0*time + h0; velocities = -g*time + v0; ``` 4. Plot the position and velocity graphs with proper labels: ```Matlab figure; subplot(2,1,1); plot(time, positions); xlabel('Time (s)'); ylabel('Height (m)'); title('Height vs Time'); subplot(2,1,2); plot(time, velocities); xlabel('Time (s)'); ylabel('Velocity (m/s)'); title('Velocity vs Time'); ``` Running the program will prompt the user for initial height and velocity, and display the height and velocity of the ball as a function of time.

Step-by-step solution

Take user inputs for initial height and velocity

Start the MATLAB program by taking the initial height and velocity of the ball as inputs from the user: ```Matlab h0 = input('Enter the initial height of the ball in meters: '); v0 = input('Enter the initial vertical velocity of the ball in meters per second: '); ```

Calculate time duration until the ball reaches the ground

For this step, we need to find when the ball reaches the ground, which means the height, h(t) is zero. We can find this by solving the position equation for t: $$ 0 = -\frac{1}{2}gt^2 + v_0t + h_0 $$ For solving the quadratic equation, you can use the quadratic formula. After this, we should choose the positive root to represent the time. ```Matlab g = 9.81; times = roots([(-1/2)*g, v0, h0]); t_final = max(times); ```

Calculate the position and velocity at different time intervals

In this step, we will create a time vector with values from 0 to t_final, and then use the position and velocity equations to calculate the ball's position and velocity at each time step. ```Matlab time = linspace(0, t_final, 1000); % 1000 time steps from 0 to t_final positions = (-1/2)*g*time.^2 + v0*time + h0; % Calculate the heights at each time velocities = -g*time + v0; % Calculate the velocities at each time ```

Plot the position and velocity graphs with proper labels

Finally, we will plot the height and velocity of the ball as a function of time using the MATLAB plot function. Also, add proper labels to the plots. ```Matlab figure; subplot(2,1,1); plot(time, positions); xlabel('Time (s)'); ylabel('Height (m)'); title('Height vs Time'); subplot(2,1,2); plot(time, velocities); xlabel('Time (s)'); ylabel('Velocity (m/s)'); title('Velocity vs Time'); ``` Now the MATLAB program is complete. Running the program will prompt the user for initial height and velocity, and then it will plot the height and velocity of the ball as a function of time.

Key concepts

Kinematic Equations

The foundational principle of mechanics, kinematic equations enable us to predict the future position and velocity of an object moving under constant acceleration. These equations connect time, displacement, initial and final velocities, and acceleration in a single-dimensional motion.

In the context of the given problem, we are dealing with a ball released with an initial velocity from a certain height. By employing the kinematic equations:

• Position: \( h(t)=\frac{1}{2} g t^{2} + v_{0} t + h_{0} \)
• Velocity: \( v(t)=g t + v_{0} \)

We can ascertain the ball's height \(h(t)\) and vertical velocity \(v(t)\) at any time point \(t\) while it is in motion. These formulas reflect the effects of Earth's gravitational acceleration (\(g\)) on the object, which is considered as a constant \(-9.81 \text{ m/s}^2\) directed downwards (negative value signifies direction opposite to the initial motion). These kinematic equations are pillars for countless applications, ranging from simple free-fall motion to providing insights into aerospace trajectories.

Programming for Engineers

Engineering problems often require careful analysis and the ability to rapidly prototype and visualize solutions. Programming languages such as MATLAB are tailored to meet these needs, providing a powerful platform for numerical computation, algorithm development, and data visualization.

For engineers, programming is not merely writing code; it's about creating clear, maintainable, and efficient solutions to complex problems. MATLAB’s syntax and workspace are specifically designed to handle mathematical tasks with ease. This is evident in the way initial conditions are inputted, how we solve for time using the quadratic formula within the software, and the seamless plotting functionality it offers. Understanding programming principles, data structures, control flows, and input/output operations are crucial components of engineering education. MATLAB’s environment fosters learning these concepts, reinforcing engineering intuition that extends beyond the screen and into real-world applications.

Plotting Graphs in MATLAB

One of MATLAB's strengths is its plotting capabilities, allowing users to visualize data intuitively and interactively. By plotting graphs, we can analyze trends, spot anomalies, and communicate results effectively.

In the provided solution, the 'plot' function is used to graph the position and velocity of the ball versus time. MATLAB allows customization of these plots in various ways to improve readability, such as adding labels, titles, and using subplots to compare multiple graphs within the same window. This visualization capability aids students and engineers alike in better understanding the behavior of systems and effectively sharing their findings. Plotting in MATLAB serves as a bridge between abstract equations and tangible understanding, enabling us to perceive nuances in data that could be missed in tabular analysis.

Quadratic Formula Calculation

A quintessential component of algebra, the quadratic formula is used to find the roots of a quadratic equation of the form \( ax^2 + bx + c = 0 \). The formal solution to this equation is given by the formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In our exercise, we apply the quadratic formula to determine the time at which the ball hits the ground. The motion of the ball dictates a quadratic relationship between height and time, which upon reaching the ground (\( h(t) = 0 \)), gives us a quadratic equation to solve.The MATLAB function 'roots' conveniently computes the roots of a polynomial equation, obviating the need to manually apply the quadratic formula, and providing a quick way to find the time at which the ball will reach the ground. It's a brilliant example of how programming can simplify traditional mathematical methods, bringing efficiency to solving engineering problems.