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# Lab 4: Projectile Motion ## Objective - To measure the range of a projectile launched at different angles and initial velocities. - To compare the experimental results with the theoretical predictions from the projectile motion model. ## Theory ### Projectile Motion Projectile motion is the m...

# Lab 4: Projectile Motion ## Objective - To measure the range of a projectile launched at different angles and initial velocities. - To compare the experimental results with the theoretical predictions from the projectile motion model. ## Theory ### Projectile Motion Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The path of the object is called its trajectory. The range (R) of a projectile, which is the horizontal distance it travels before hitting the ground, depends on the initial velocity ($v_0$), the launch angle ($\theta$), and the acceleration due to gravity (g). Assuming no air resistance and a flat surface, the range R can be calculated using the following formula: $R = \frac{v_0^2 \sin(2\theta)}{g}$ Where: - $v_0$ is the initial velocity of the projectile. - $\theta$ is the launch angle with respect to the horizontal. - g is the acceleration due to gravity ($9.8 m/s^2$). ### Key Concepts to Consider: 1. **Independence of Motion**: The vertical and horizontal components of motion are independent of each other. 2. **Vertical Motion**: Subject to constant acceleration due to gravity. 3. **Horizontal Motion**: Constant velocity (assuming no air resistance). 4. **Launch Angle**: The angle at which the projectile is launched significantly affects both the range and maximum height. ## Materials and Equipment - Projectile launcher - Steel ball - Measuring tape - Protractor - Carbon paper - White paper - Target board ## Procedure 1. **Set Up**: - Set up the projectile launcher on a level surface. - Adjust the launcher to a specific angle using the protractor. 2. **Measuring Initial Velocity**: - Use the projectile launcher to shoot the steel ball horizontally. - Measure the vertical distance (y) from the launch point to the ground and the horizontal distance (x) the ball travels before hitting the ground. - Calculate the initial velocity ($v_0$) using kinematics equations: $v_0 = x \sqrt{\frac{g}{2y}}$ 3. **Range Measurement**: - Set the launcher to a predetermined angle. - Fire the steel ball multiple times (e.g., 5 times) at the same angle and initial velocity. - Use carbon paper on the target to mark the landing spot of the ball. - Measure the horizontal distance from the launch point to each landing spot. - Calculate the average range for each angle. 4. **Varying the Angle**: - Repeat the range measurement for different launch angles (e. g., 30°, 45°, 60°). - Keep the initial velocity constant. 5. **Varying Initial Velocity**: - Set the launcher to a fixed angle. - Adjust the launcher to vary the initial velocity (if possible). - Measure the range for different initial velocities. 6. **Data Recording**: - Record all measurements, including launch angles, initial velocities, and ranges, in a table. ## Data Analysis 1. **Calculate Theoretical Range**: - Using the measured initial velocities and launch angles, calculate the theoretical range using the formula: $R = \frac{v_0^2 \sin(2\theta)}{g}$ 2. **Compare Experimental and Theoretical Results**: - Compare the experimental ranges with the theoretical ranges. - Calculate the percent difference between the experimental and theoretical ranges using the formula: $\text{Percent Difference} = |\frac{\text{Experimental Range - Theoretical Range}}{\text{Theoretical Range}}| \times 100%$ 3. **Graphs**: - Plot graphs of range vs. launch angle for both experimental and theoretical values. - Plot graphs of range vs. initial velocity for both experimental and theoretical values. ## Results Record your measurements and calculations in the following tables: ### Table 1: Measuring Initial Velocity | Trial | Vertical Distance y (m) | Horizontal Distance x (m) | Initial Velocity $v_0$ (m/s) | | :---- | :----------------------- | :------------------------ | :----------------------------- | | 1 | | | | | 2 | | | | | 3 | | | | | 4 | | | | | 5 | | | | | Average | | | | ### Table 2: Range vs. Launch Angle (Constant Initial Velocity) Initial Velocity, $v_0 = $ _________ m/s | Launch Angle $\theta$ (degrees) | Experimental Range (m) | Theoretical Range(m) | Percent Difference (%) | | :----------------------------- | :----------------------- | :----------------------- | :--------------------- | | 30° | | | | | 45° | | | | | 60° | | | | ### Table 3: Range vs. Initial Velocity (Constant Launch Angle) Launch Angle, $\theta = $_________ degrees | Initial Velocity $v_0$(m/s) | Experimental Range (m) | Theoretical Range (m) | Percent Difference (%) | | :----------------------------- | :----------------------- | :----------------------- | :--------------------- | | | | | | | | | | | | | | | | ## Discussion - Discuss the results obtained. - Explain any discrepancies between experimental and theoretical values. - Discuss possible sources of error, such as air resistance, measurement errors, and inconsistencies in the launch mechanism. - Discuss the effect of launch angle and initial velocity on the range of the projectile. - Compare your results with what you expected based on the theory of projectile motion. ## Conclusion - Summarize the main findings of the experiment. - State whether the objective of the experiment was achieved. - Suggest possible improvements to the experiment.