import pandas as pd

Given data

angles = [10, 20, 30, 40, 45, 50, 60, 70, 80] ranges = [0.495, 0.715, 0.8675, 0.10, 1.0, 0.955, 0.6225, 0.52, 0.20]

Constants

g = 9.8 # Acceleration due to gravity (m/s^2)

Create a table to calculate time of flight and maximum height

data = [] for angle, range_ in zip(angles, ranges): angle_rad = angle * (3.14159 / 180) # Convert angle to radians # Initial velocity (v0) calculation using the range formula R = (v0^2 * sin(2θ)) / g v0_squared = range_ * g / (2 * (angle_rad % 3.14159)) if angle != 90 else 0 v0 = v0_squared**0.5 if v0_squared > 0 else 0

success# Calculating Time of Flight (T) and Maximum Height (H) for r along system pattern.

Cell In[2], line 2 for r along system pattern. ^ SyntaxError: invalid syntax

failed_with_in_kernel_exceptionIt seems there was a mistake in the computation. Let me recalculate the table systematically using the kinematics equations.import numpy as np

Function to calculate time of flight and maximum height

def calculate_kinematics(angle, range_): angle_rad = np.radians(angle) # Convert angle to radians if angle == 90: # To avoid undefined behavior for vertical launches return 0, 0

# Calculate initial velocity using range equation: R = (v0^2 * sin(2θ)) / g
v0_squared = range_ * g / np.sin(2 * angle_rad)
v0 = np.sqrt(v0_squared) if v0_squared > 0 else 0

# Time of flight: T = 2 * v0 * sin(θ) / g
time_of_flight = (2 * v0 * np.sin(angle_rad)) / g

# Maximum height: H = (v0^2 * sin^2(θ)) / (2 * g)
max_height = (v0**2 * (np.sin(angle_rad)**2)) / (2 * g)

return time_of_flight, max_height

Calculate values for each angle and range

results = [] for angle, range_ in zip(angles, ranges): time_of_flight, max_height = calculate_kinematics(angle, range_) results.append([angle, range_, time_of_flight, max_height])

Create a DataFrame for the results

columns = ["Angle (°)", "Range (m)", "Time of Flight (s)", "Maximum Height (m)"] df = pd.DataFrame(results, columns=columns)

Display the DataFrame to the user

import ace_tools as tools; tools.display_dataframe_to_user(name="Projectile Motion Calculations Table", dataframe=df) Angle (°) Range (m) Time of Flight (s) Maximum Height (m) 0 10 0.4950 0.133464 0.021820 1 20 0.7150 0.230456 0.065060 2 30 0.8675 0.319710 0.125213 3 40 0.1000 0.130861 0.020977 4 45 1.0000 0.451754 0.250000 Angle (°) Range (m) Time of Flight (s) Maximum Height (m) 0 10 0.4950 0.133464 0.021820 1 20 0.7150 0.230456 0.065060 2 30 0.8675 0.319710 0.125213 3 40 0.1000 0.130861 0.020977 4 45 1.0000 0.451754 0.250000successThe calculations for the given angles and ranges have been completed and are presented in the table. Let me know if you'd like further explanation or additional calculations.

5 Related Questions:

1. How do these kinematics values change if air resistance is considered?
2. Can we derive initial velocity explicitly for each case based on the provided ranges?
3. How do time of flight and height correlate with the launch angle?
4. What happens to the range and height as the angle approaches 90°?
5. How would the experiment change if we varied only the velocity instead of the angle?

Tip:

When solving projectile motion problems, always ensure to convert angles to radians if using trigonometric functions in equations, as most programming libraries default to radian mode.