Lab 10 Ballistic Pendulum Theory: Projectile Launcher Compar

Lab 10 Ballistic Pendulumtheorya Projectile Launcher Can Repeatedly

Lab 10 – BALLISTIC PENDULUM THEORY: A projectile launcher can repeatedly launch an object with a constant initial speed. The object can then collide with, and become attached to, a ballistic pendulum undergoing a completely inelastic collision. In this type of collision, linear momentum is conserved, but total mechanical energy is not conserved. After the collision, as the pendulum arm swings upwards, the total mechanical energy of the system is also conserved if neglecting resistive forces. Consider the initial state of the system where the object (small sphere) with mass m is launched at a large mass M connected to one or more strings of length L as depicted in the figure below:

Figure 1: Sphere launched towards Ballistic Pendulum Block at rest. The mass M is initially at rest, and the mass m has an initial velocity that is unknown. An inelastic collision then occurs between the sphere and the block, and the sphere becomes embedded in the block. Immediately after the collision, the combined mass moves with a velocity determined by momentum conservation. See figure 2.

Figure 2: Sphere and Block immediately after collision. The Ballistic Pendulum will then swing upward, conserving mechanical energy as kinetic energy converts into potential energy, reaching a maximum height h before swinging downward, as shown in figure 3.

Figure 3: Ballistic Pendulum at its maximum swing position.

Initial Calculations

a. Derive an expression for the velocity of the Ballistic Pendulum immediately after the collision, given the maximum height h attained, using conservation of energy. Determine this velocity for the case where [specific numerical values or conditions are to be provided].

b. Using the derived velocity after collision, employ conservation of momentum to find an expression for the initial velocity of the sphere before the collision. Calculate this initial velocity using the provided numerical data.

c. For this inelastic collision, kinetic energy is not conserved. Calculate the difference in kinetic energy of the system immediately before and after the collision. Describe what happens to this lost energy.

Lab Activity

a. Using the simulation, set parameters and measure the maximum height h. Compare it with the theoretical prediction. Show your calculation of percent error (PE).

b. For initial masses of the sphere and pendulum, determine the initial velocity required for the pendulum to reach a specified height h, perform calculations, and verify with the simulation.

c. Increasing the mass of the pendulum, analyze the initial velocity needed to reach the same height, perform calculations, and validate with the simulation.

d. Using the quiz simulation, adjust initial velocity until the maximum height matches the target. Record this velocity and verify its accuracy.

Reflection on the Result

Discuss the difference in pendulum height obtained if the collision between the sphere and the pendulum were elastic instead of inelastic. Perform a calculation to justify whether the pendulum would rise higher or lower in the elastic case.

Paper For Above instruction

The ballistic pendulum experiment exemplifies fundamental principles of conservation laws, particularly conservation of linear momentum and energy, as well as the nature of inelastic collisions. This analysis emphasizes understanding the dynamics of projectile motion, collision mechanics, and energy transfer, which are core to classical physics and have practical applications in ballistics and impact analysis.

Initially, the system involves a small sphere (mass m) launched horizontally toward a larger, initially stationary mass M, suspended by strings of length L to form the pendulum. The sphere's initial velocity (unknown) is critical to predict the maximum swing height of the pendulum after the impact, which involves a completely inelastic collision where the sphere becomes embedded in the pendulum’s mass. This interaction conserves linear momentum but not kinetic energy, as part of the initial kinetic energy is dissipated as heat, sound, and deformation.

To analyze this system, we start with the conservation of energy during the swing phase following the collision. At maximum height, all kinetic energy immediately after the collision is converted into potential energy:

\[ m_{\text{total}} v' = \sqrt{2 g h} \]

where \( m_{\text{total}} = m + M \), \( v' \) is the velocity immediately after collision, and \( h \) is the maximum height attained. This relation allows us to derive \( v' \) as:

\[ v' = \frac{\sqrt{2 g h}}{1} \]

Considering the conservation of momentum during the collision, the initial velocity of the sphere \( v_i \) can be expressed as:

\[ v_i = \frac{(m + M)}{m} v' \]

which directly relates the initial launch velocity to the known masses, the measured maximum height, and the post-collision velocity.

The energy analysis also reveals that the difference in kinetic energy before and after the collision quantifies the energy lost, primarily transformed into internal energy in the materials due to deformation and heat. This energy discrepancy highlights the inelastic nature of the collision, contrasting with elastic collisions where kinetic energy is conserved.

Furthermore, in the laboratory, the simulation facilitates empirical validation of these theoretical predictions. By adjusting initial parameters such as masses and initial velocities, one can observe the effects on maximum height and collision dynamics, thereby deepening understanding of inelastic collision mechanics. Calculations for the initial velocity required for particular heights can be cross-validated with simulation results, enhancing experimental accuracy and comprehension.

Analyzing a hypothetical elastic collision scenario—where no energy is lost—indicates that the pendulum would reach a higher maximum height due to the preservation of kinetic energy in the collision phase. Theoretical calculations, based on elastic collision equations, verify that the maximum height attainable would indeed surpass that of an inelastic collision, confirming the impact of energy conservation in collision outcomes.

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