Question

Consider the simplest program for evaluting the formula \(y(t)=v_{0} t-\) \(0.5 g t^{2}:\) $$ \begin{aligned} &v 0=3 ; g=9.81 ; t=0.6 \\ &y=v 0 * t-0.5 * g * t * 2 \\ &\text { print } y \end{aligned} $$ Modify this code so that the program asks the user questions \(t=?\) and v0 \(=\) ?, and then gets \(t\) and vo from the user's input through the keyboard. Name of program file: ball_qa.py. \(\diamond\)

Short answer

Modify the code to use `input()` for \( t \) and \( v_0 \), then calculate and print \( y(t) \).

Step-by-step solution

Understanding the Formula

The given formula represents the height of an object at time \( t \), with initial velocity \( v_0 \) and acceleration due to gravity \( g \). It is expressed as:\[y(t) = v_0 t - 0.5 g t^2\]where \( g \) is programmed as 9.81 (the acceleration due to gravity in m/s²).

Identify Input Variables

The exercise requires us to modify the code so that \( t \) (time) and \( v_0 \) (initial velocity) are provided by the user. Currently, they are set as constants in the code.

Modify the Code for User Input

To allow users to input the values for \( t \) and \( v_0 \), we will use the `input()` function in Python. This function will prompt the user for input and return it as a string, which we'll convert into a float for calculation purposes.

Implement the Modified Code

Here is the modified Python code for the program:```python# Prompt user for input and convert to floatt = float(input('t=? '))v0 = float(input('v0=? '))# Given constantg = 9.81# Calculate yy = v0 * t - 0.5 * g * t**2# Print resultprint(y)```This program will now ask the user for the time \( t \) and initial velocity \( v_0 \), perform the calculation, and print the height \( y \).

Save and Test the Program

Save the modified code in a file named `ball_qa.py`. Run the program in a Python environment to ensure it asks for user inputs and computes the result accurately.

Key concepts

User Input in Python Programming

Interacting with users through input is a fundamental aspect of Python programming. It enhances the flexibility and usability of a program by allowing users to provide custom data or preferences for execution.

In the context of this exercise, we aim to prompt users for time \(t\) and initial velocity \(v_0\). The function `input()` is a simple yet powerful tool for this purpose. When invoked, `input()` displays a prompt, waits for user input, and reads the input as a string. However, since we need numerical values for physics computations, the input must be converted to a floating-point number using `float()`. This allows the program to handle decimals, crucial for precise calculations.

Using `input()`, we can dynamically replace static values like \(t\) and \(v_0\) in our formula with user-provided data. This modification is encapsulated in the Python code:

• `t = float(input('t=? '))`
• `v0 = float(input('v0=? '))`

By executing these lines, the program efficiently transitions from static calculations to a user-interactive model.

Physics Computation of Projectile Motion

The formula \[y(t) = v_0 t - 0.5 g t^2\] is used in physics to compute the vertical position of a projectile at time \(t\). Understanding this formula involves recognizing the roles of each component.

Here:

• \(v_0\) is the initial velocity, determining how fast and in what direction the object starts moving.
• \(g\) is the acceleration due to gravity, a constant at approximately \(9.81 \, \text{m/s}^2\), influencing how quickly the object slows down or speeds up in the vertical trajectory.
• \(t\) is the time elapsed, indicating how long it has been since the object started moving.

The formula derives from the basic principles of kinematics, where the first term, \(v_0 t\), gives the initial linear motion, and the second term, \(0.5 g t^2\), accounts for the effects of gravity over time.

This computation allows us to predict where and when an object will reach a certain height, an essential aspect of analyzing motion in physics.

Program Modification for Dynamic Inputs

Modifying a program to handle dynamic inputs involves several steps that showcase the versatility of programming languages like Python. It enables a single script to perform a variety of calculations based on user needs, rather than being confined to predefined values.

The key changes required for the current task involve harnessing Python's ability to read and convert user inputs, incorporating:

• Replacement of constant values with the `input()` function to gather data directly from the user. This makes the script interactive and adaptable to different scenarios.
• Conversion of strings to floating-point numbers through `float()`, ensuring compatibility with the arithmetic operations needed for accurate physics computation.
• Designing a user-friendly interaction by carefully worded prompts like `'t=? '` and `'v0=? '`, making it clear what information the user needs to provide.

These modifications not only tailor the program to user specifications but also enhance learning by offering a hands-on experience in coding and physics calculations. As a final step, always test the program to verify its functionality and correctness, ensuring it meets the expectations.