Examine Collisions With Two Blocks: Momentum And Elasticity ✓ Solved

Examine collisions with two blocks: momentum and elasticity.

We are going to examine what happens during collisions in a simple 1D two-block momentum-simulation. The program uses elasticity to govern how elastic (energy-conserving) the collision is, with e=1 being elastic, e=0 being completely inelastic, and values in between representing partial elasticity. Set up the simulation with equal masses and the velocity of one block set to 0, and elasticity e=0. What do you expect to happen?

What happens? Try changing the masses. What do you observe? Pause the simulation after the collision to read off the values. Does it act as you expect?

Try changing the initial velocities. Be sure to try with both masses going in the same direction initially (but make sure they actually collide). Also try equal and opposite velocities. How is the motion affected?

Reset the simulation to equal masses and one initial velocity zero, but set elasticity to 1. This creates a perfectly elastic collision. The equations for elastic collisions are complicated to derive. You can look these up if you need them. Using these equations, what do you expect to happen? Try giving them differing initial velocities. Pause the simulation after the collision so you can read off the values. Be sure to try with both masses going in the same direction initially (but make sure they actually collide). Also try equal and opposite velocities. Does it act as you expect? Try making the blocks unequal mass. What do you expect to happen? If you try it in the simulation, what do you see? Try different combinations of velocities.

Reset the simulation to the default values, but set elasticity to 0.5. This creates an inelastic collision, the most general type. For inelastic collisions, we cannot find both speeds after the collision, we can only find one speed after the collision once we know the other. Try several combinations of initial velocities and masses, and comment on what you see. Be sure to try with both masses going in the same direction initially (but make sure they actually collide). Although we cannot get an exact equation, you should be able to see some patterns.

Paper For Above Instructions

In this study, we analyze a classic one-dimensional collision scenario involving two blocks with adjustable masses, initial velocities, and a tunable elasticity parameter e, which ranges from 0 (perfectly inelastic) to 1 (perfectly elastic). The problem is an idealized but pedagogically powerful system to illustrate momentum conservation, kinetic energy exchange, and the role of the coefficient of restitution in determining post-collision velocities. Throughout, we assume a single axis of motion and no external forces (e.g., friction or external impulses) acting during the collision.

1) Theoretical framework. Collisions in one dimension conserve total linear momentum, regardless of the elasticity, provided there are no external impulses. If m1 and m2 are the masses and u1 and u2 their pre-collision velocities, the momentum before and after collision satisfies m1 u1 + m2 u2 = m1 v1 + m2 v2, where v1 and v2 are the post-collision velocities (Halliday, Resnick, & Walker, 2014). The coefficient of restitution e is defined by the relation v2 − v1 = − e (u2 − u1). For one-dimensional collisions, the exact post-collision velocities for arbitrary e are given by v1 = [(m1 − e m2) u1 + (1 + e) m2 u2] / (m1 + m2) and v2 = [(m2 − e m1) u2 + (1 + e) m1 u1] / (m1 + m2) (Goldstein, Poole, & Safko, 2002). The energy lost to deformation and internal degrees of freedom increases as e decreases from 1 toward 0 (Kleppner & Kolenkow, 2013). These equations illustrate how momentum and energy transfer depend on mass ratio, initial speeds, and elasticity.

2) Case: equal masses, one initially at rest, e = 0. When m1 = m2 and u2 = 0, a perfectly inelastic collision (e = 0) causes the blocks to move together after impact with a common speed v = u1 / 2. Momentum is conserved, but kinetic energy is not; half of the initial kinetic energy is typically lost to deformation and internal modes (Serway & Jewett, 2018). This is a concrete demonstration of energy dissipation in real collisions and helps students connect abstract formulas to observable behavior (Halliday et al., 2014).

3) Case: equal masses, elastic collision e = 1. With m1 = m2 and e = 1, the blocks exchange velocities: v1 = u2 and v2 = u1. If one mass starts at rest, the moving mass stops and transfers all its velocity to the other block. Momentum remains constant and kinetic energy is fully recovered in one dimension (Giancoli, 2013). This idealized outcome—perfect energy restitution and velocity exchange—highlights the distinct difference between elastic and inelastic cases (Tipler & Mosca, 2007).

4) Case: unequal masses, elastic e = 1. For general m1 ≠ m2, the post-collision velocities depend on the mass ratio and initial velocities; the total momentum remains constant and a portion of the initial kinetic energy is redistributed between the blocks. The equations show that the lighter block tends to rebound with a higher velocity when struck by a heavier block moving toward it, while the heavier block’s velocity changes more modestly (Kleppner & Kolenkow, 2013). In practice, the simulation should reveal changing rebound speeds that reflect both mass difference and the restitution coefficient (Halliday et al., 2014).

5) Case: opposite directions and same-direction approaches. When initial velocities are opposite, the relative velocity is larger, often yielding a more pronounced exchange of momentum and kinetic energy, particularly for equal masses. When both blocks approach from the same direction, a collision still conserves momentum but the resulting velocities can differ markedly depending on mass ratio and e. The qualitative behavior—whether one block overtakes the other or both reverse directions—depends on the combination of masses and e (Taylor, 2005).

6) Case: intermediate elasticity, e = 0.5. This yields partial energy restitution. You cannot generally determine both final speeds solely from energy considerations, as some kinetic energy is stored temporarily as deformation and then released. The observed final speeds will satisfy momentum conservation and reflect partial energy recovery (Fowles & Cassiday, 2005). The pattern is that higher e yields more post-collision kinetic energy and more “perfect” rebound behavior, while lower e yields greater energy loss and more inelastic, coupled motion after impact (Halliday et al., 2014).

7) Practical interpretation and study design. The simulation’s value lies in enabling students to pause after a collision, observe speeds, and compare them to analytical predictions derived from the above equations. By varying masses, initial velocities, and e, students can systematically verify momentum conservation and witness the dependence of post-collision states on material properties (coefficient of restitution) and the mass distribution (Kleppner & Kolenkow, 2013). The exercise reinforces the conceptual distinction between elastic and inelastic collisions and demonstrates the trade-offs between energy conservation and momentum transfer in real-world interactions (Giancoli, 2013).

8) Synthesis and patterns. Across scenarios, momentum conservation holds regardless of elasticity, while kinetic energy conservation is only guaranteed for e = 1. Unequal masses produce asymmetric velocity changes, and the energy lost in inelastic collisions increases as e decreases. Observing this across multiple initial conditions reveals robust patterns: exchange or redistribution of speeds aligns with the mathematical predictions, and the degree of energy loss scales with the restitution coefficient (Halliday et al., 2014; Serway & Jewett, 2018). These patterns are essential for building intuition about real-world collisions in mechanical systems and for connecting simulations to laboratory experiments (Taylor, 2005).

References

• Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics (10th ed.). Wiley.
• Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers (9th ed.). Cengage.
• Giancoli, D. (2013). Physics: Principles with Applications (7th ed.). Pearson.
• Kleppner, D., & Kolenkow, R. (2013). An Introduction to Mechanics (2nd ed.). Cambridge University Press.
• Goldstein, H., Poole, C. P., & Safko, J. (2002). Classical Mechanics (3rd ed.). Addison-Wesley.
• Taylor, J. R. (2005). Classical Mechanics. University Science Books.
• Fowles, G. L., & Cassiday, D. C. (2005). Analytical Physics (6th ed.). Thomson Brooks/Cole.
• Tipler, P. A., & Mosca, G. (2007). Physics for Scientists and Engineers (6th ed.). W. H. Freeman.
• HyperPhysics. (n.d.). Elastic collisions and coefficient of restitution. Retrieved from http://hyperphysics.phy-astr.gsu.edu/hbase/Collision/Collide.html
• Khan Academy. (n.d.). Collisions and momentum. Retrieved from https://www.khanacademy.org/science/physics/one-dimensional-motion