To calculate the time of flight and maximum height for each angle based on the given ranges, we use kinematic equations. Here's the approach:

Key Equations:

1. Range Equation: $R = \frac{v_0^2 \sin(2\theta)}{g}$ Where $R$ is the range, $v_0$ is the initial velocity, $\theta$ is the angle of projection, and $g = 9.8 \, \text{m/s}^2$ (acceleration due to gravity).

2. Time of Flight: $T = \frac{2v_0 \sin(\theta)}{g}$

3. Maximum Height: $H = \frac{(v_0 \sin(\theta))^2}{2g}$

Steps:

1. Solve for $v_0$ (initial velocity) using the range equation.
2. Calculate $T$ (time of flight) using the initial velocity and angle.
3. Calculate $H$ (maximum height) using the initial velocity and angle.

I will calculate the values for each angle.import math import pandas as pd

Given data

angles = [10, 20, 30, 40, 45, 50, 60, 70, 80] # degrees ranges = [0.495, 0.715, 0.8675, 0.1, 0.1, 0.955, 0.6225, 0.52, 0.2] # meters g = 9.8 # acceleration due to gravity in m/s^2

Function to calculate v0, time of flight (T), and max height (H)

def calculate_kinematics(angle, range_): angle_rad = math.radians(angle) # Calculate initial velocity v0 from range equation v0 = math.sqrt((range_ * g) / math.sin(2 * angle_rad)) # Time of flight T T = (2 * v0 * math.sin(angle_rad)) / g # Maximum height H H = (v0 * math.sin(angle_rad))**2 / (2 * g) return v0, T, H

Perform calculations for each angle and range

results = [] for angle, range_ in zip(angles, ranges): v0, T, H = calculate_kinematics(angle, range_) results.append({"Angle (°)": angle, "Range (m)": range_, "Initial Velocity (m/s)": v0, "Time of Flight (s)": T, "Max Height (m)": H})

Convert results to DataFrame for better visualization

results_df = pd.DataFrame(results)

Display the results to the user

import ace_tools as tools; tools.display_dataframe_to_user(name="Projectile Motion Analysis", dataframe=results_df) Angle (°) Range (m) Initial Velocity (m/s) Time of Flight (s)
0 10 0.4950 3.766082 0.133464
1 20 0.7150 3.301660 0.230456
2 30 0.8675 3.133159 0.319710
3 40 0.1000 0.997556 0.130861
4 45 0.1000 0.989949 0.142857

Max Height (m)
0 0.021820
1 0.065060
2 0.125213
3 0.020977
4 0.025000 Angle (°) Range (m) Initial Velocity (m/s) Time of Flight (s)
0 10 0.4950 3.766082 0.133464
1 20 0.7150 3.301660 0.230456
2 30 0.8675 3.133159 0.319710
3 40 0.1000 0.997556 0.130861
4 45 0.1000 0.989949 0.142857

Max Height (m)
0 0.021820
1 0.065060
2 0.125213
3 0.020977
4 0.025000 successThe table containing the calculated values for initial velocity, time of flight, and maximum height for each angle has been provided. Let me know if you need further explanations or additional calculations.

Here are some related questions to deepen your understanding:

1. How does the angle of launch affect the maximum height?
2. Why is the range maximum at a specific angle (approximately 45°)?
3. Can you explain the relationship between initial velocity and maximum height?
4. How would air resistance alter these results?
5. How could you experimentally verify these calculations?

Tip: Always ensure the units are consistent in kinematic equations to avoid calculation errors.