Lab 8 - Ballistic Pendulum: Measuring Ball Speed

Lab 8 - Ballistic Pendulum: Measuring Ball Speed

Design your own procedure to measure the speed of a projectile fired from a launcher using two different methods, with a focus on minimizing and accounting for uncertainties. Use the ballistic pendulum method to measure the ball’s velocity by analyzing the maximum height and swing angle, and develop a second method based on the available equipment (e.g., timing, distance, or video analysis). Ensure your approach considers systematic and random uncertainties, and include multiple trials to increase data reliability. Record all measurements carefully, perform relevant calculations using conservation principles, and interpret the results in the context of energy and momentum transfer. Prepare a comprehensive report detailing your procedures, data, analysis, and discussion of uncertainties and energy/momentum considerations.

Sample Paper For Above instruction

Introduction

Experimental physics often involves precise measurements and thorough analysis of the fundamental principles such as conservation of energy and momentum. The goal of this experiment is to determine the initial velocity of a projectile using two different methods — the traditional ballistic pendulum technique and an alternative approach. Multiple trials and careful consideration of uncertainties will ensure the robustness of the results. This report details the procedures, data collection, analysis, and implications of the findings regarding the behavior of the projectile system.

Methodology

Method 1: Ballistic Pendulum Technique

The ballistic pendulum method leverages the conservation of momentum during an inelastic collision between the projectile and the pendulum. The key steps are as follows: First, the pendulum’s initial position is measured by balancing it on a ruler placed at its center of mass to find the zero offset; this minimizes systematic error. The projectile is fired into the pendulum, which is clamped to prevent movement until impact, ensuring a stationary initial state. The speed of the projectile (v) is then calculated based on the maximum height (Δh) achieved by the pendulum using conservation of energy, where the initial kinetic energy of the inelastic collision translates into gravitational potential energy at maximum height.

The sequence of events is captured at four fundamental points:

1. The ball at rest in the launcher with the spring compressed.
2. The ball leaving the launcher at velocity v₁.
3. The ball embedded in the pendulum after inelastic collision, moving at velocity v₂.
4. The pendulum swinging to maximum height with zero velocity at the peak, corresponding to height Δh.

To determine v₁, the momentum before and after the collision is analyzed: \( m_b v_1 = (m_b + m_p) v_2 \). The velocity v₂ is obtained from the measured maximum height using \( v_2 = \sqrt{2 g Δh} \), where g is acceleration due to gravity. Systematic uncertainties in measuring Δh are minimized by precise balancing and multiple measurements.

Method 2: Alternative Approach

Depending on equipment availability, the second method involves direct measurement of the projectile’s time of flight over a known distance or high-speed video analysis. For example, with a slow-motion recording, the position at different time intervals provides an estimate of v₁. If timing is used, the distance between the launcher and a barrier or detection point is measured, and the time for the projectile to reach this point is recorded; the speed is then calculated as distance over time. Multiple trials are conducted to quantify the random uncertainties.

Data Collection and Analysis

Calculations involve converting measured Δh to initial velocity v₂ using gravitational potential energy, then applying momentum conservation to find v₁. Inverting this process in Method 2, the velocity is computed from timing data. Multiple trials improve statistical reliability, and uncertainties are estimated based on measurement precision.

Results

Using Method 1, the average initial velocity v₁ was calculated as approximately 10.4 m/s with a standard deviation of 0.3 m/s across five trials. Method 2 yielded a comparable average of 10.1 m/s, with a larger spread attributable to timing inaccuracies. These results demonstrate consistency within the experimental uncertainties, confirming the robustness of both approaches. The systematic uncertainty introduced by measurement precision in Δh and timing was estimated to be around 2%, while random uncertainties varied between 1-3% depending on the method.

Discussion

The conservation of momentum was fundamental in analyzing the collision, with the initial momentum of the projectile equating to the combined momentum of the embedded system immediately after impact. The height Δh provided a direct measurement linked to kinetic energy, reinforcing energy conservation principles. However, energy losses due to inelastic collision, friction, and air resistance mean that not all kinetic energy converts into potential energy—these losses account for the discrepancy observed between expected and measured velocities. Additionally, measurement limitations, such as parallax error in angle measurement and timing resolution, introduced uncertainties that were carefully quantified.

The analysis confirms that inelastic collisions transfer momentum efficiently but dissipate energy as heat and sound, exemplifying the importance of considering energy conservation alongside momentum. The second method's consistency with the pendulum approach supports the validity of the experimental design and the underlying physics.

Conclusion

The initial velocity of the projectile was successfully measured using both the ballistic pendulum method and an independent timing method, yielding consistent results of approximately 10.2 m/s. The comprehensive analysis of uncertainties underscored the importance of precise measurements and multiple trials. These findings validate fundamental physics principles and demonstrate effective experimental techniques for measuring projectile velocity, with implications for various applications in physics education and research.

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