This experiment appears to involve projectile motion, specifically launching a dart at different angles and velocities, and analyzing the resulting range (distance traveled) and time of flight. Here’s how you can proceed with setting up and performing the experiment, followed by an explanation of the data analysis.

Steps for the Experiment

1. Set Up the Dart Pipe:

   • Set the pipe at three different angles: 30°, 45°, and 60° relative to the ground.
   • Ensure that the setup is stable, with a consistent launching mechanism.

2. Launch the Dart:

   • Launch the dart from the pipe for each angle.
   • Use a consistent force or launching mechanism to maintain initial velocity for each set of trials.

3. Measure the Distance:

   • After launching the dart, measure the horizontal distance (range) from the launching point to where the dart lands.
   • Use a tape measure for accuracy.

4. Record the Time of Flight:

   • Using a stopwatch, record the time taken for the dart to hit the ground after it is launched.

5. Repeat for Each Angle and Velocity:

   • Perform the experiment for each angle (30°, 45°, 60°).
   • Vary the initial velocity (e.g., by adjusting the launching force) and repeat the trials at least 3-5 times for each angle and velocity combination to get reliable data.
   • Record the distance and time for each trial.

6. Plot the Data:

   • For each launch angle, plot the distance (range) on the y-axis and the launch angle on the x-axis.
   • You can also plot a second graph with time of flight on the y-axis and the launch angle on the x-axis.
   • Analyze the trends in both graphs to determine how the launch angle affects the range and flight time.

Analysis and Interpretation

1. Projectile Motion Equations: The range (R) of a projectile launched at an angle $\theta$ with initial velocity $v_0$ is given by the equation: $R = \frac{v_0^2 \sin(2\theta)}{g}$ where $g$ is the acceleration due to gravity (9.8 m/s²).

2. Time of Flight: The time of flight (T) can be calculated using: $T = \frac{2v_0 \sin(\theta)}{g}$

3. Expected Observations:

   • The range will generally increase with angle up to a point (usually around 45°), after which it will start to decrease.
   • The time of flight should increase with the angle because a higher angle results in a longer vertical component of the motion.

4. Graph Analysis:

   • The range vs. angle plot should peak at 45°, showing the longest range for this angle.
   • The time vs. angle plot will increase as the launch angle increases since the vertical motion lasts longer at higher angles.

Would you like details on how to improve accuracy or have questions about any specific steps?

Follow-Up Questions:

1. How does changing the initial velocity affect the range for a fixed launch angle?
2. Why is 45° considered the optimal angle for maximum range?
3. How does air resistance factor into real-world results, and how would it change your calculations?
4. Can you calculate the theoretical range and time of flight for each angle using the equations provided?
5. How could you measure the velocity more accurately to improve your experimental results?

Tip: When repeating the experiment, try to launch the dart at least three times for each setting to ensure consistency in your results.