Pre-Lab Questions: Billiard Ball Collides Head-On

Pre Lab Questions1 A Billiard Ball Collides Head On With Billiard Bal

Pre-Lab Questions 1. A billiard ball collides head-on with billiard ball at rest. a. Sketch a position vs. time graph for each ball, include position before the collision, once the collision occurs and after the collision. b. What can be said about the conservation of momentum for the collision? Explain your answer.

2. Write down the total momentum for two marbles of mass, m, both moving at velocity, v. What is the kinetic energy of the system? 3. When you drop two marbles at once, why doesn’t only one marble come off the end twice as fast? Write down the kinetic energy of one marble with mass m and velocity 2v and compare this to your answer in Experiment 1 Post Lab Question 4 to check. Note: Assume the collisions are perfectly elastic. © 2014 eScience Labs, LLC. All Rights Reserved NATS 1750 A (Winter 2017): Assignment 3, Version 1.0 - February 20, 2017 Due: March 15, 2017 by 11 pm EDT via Moodle (Late Penalty: 25% per day - including weekends. Strictly enforced.) Instructions: â— You are expected to provide answers for ​every​ question. You are encouraged to show all of your work so that marks can be awarded for partially correct answers. â— Although you are encouraged to collaborate with your classmates, each of you is expected to submit a separate and distinct assignment - a point that will require acknowledgement upon submission.

1. Using Google Maps (or a similar mapping capability): a. Provide the latitude and longitude for a feature of your choosing on Mount Royal. [2 marks] b. Using your map, or some other source, provide an estimate for the altitude of Mount Royal. (Be certain to cite your source of this estimate.) [1 mark] c. Using your map, or some other source, provide an estimate for the altitude of area around Mount Royal. (Be certain to cite your source of this estimate.) [1 mark] d. Using your map, provide a three-dimensional picture of Mount Royal that illustrates its ​relief​ relative to the surrounding area. Based on your answers 1(b) and 1(c), provide a quantitative estimate for the relief of Mount Royal. [4 marks] e. Based on your answer to 1(d), estimate the minimum depth of rock that the Mount Royal would have required for intrusion (as an igneous pluton - i.e., assume Mount Royal intruded into a preexisting strata of sedimentary rocks.) [2 marks] f. Through use of a sketch, provide a geological reconstruction for Mount Royal around the time at which the intrusion took place. Add sketches that provide before and after illustrations. [3 marks each = 9 marks; add 3 bonus marks for 3D block diagrams] g. Provide a geologically sound narrative to account for Mount Royal’s evolution through use of your sketches - from the past, through to how it appears today. [3 marks] © L. I. Lumb - Sharing prohibited. Violators subject to legal and/or academic consequences. 1 Locate a geological map of Mount Royal (e.g., ​Plandowski's​ or ​Eby’s Figure 3​). a. Prepare a geological cross section that trends East-West. Your cross section should be to scale, and intersect with the thin Utica shale lithology that is present at the easternmost edge of Mount Royal. Label each lithology on your cross section, and indicate which fundamental rock type it represents. In addition to your cross section, provide an annotated version of the geological map you used to indicate the location of your cross section in map view. [10 marks] b. According to your cross section, which lithology (or lithologies) has the Mount Royal pluton intruded into? [2 marks] c. From your geological section, estimate the thickness of the Utica shale lithology at the easternmost edge of the pluton. [3 marks] d. According to Bodycomb (2001), the metamorphic aureole extended for about 800 m. Indicate this on your cross section by extending an 800 m thick region from the edge of the last igneous lithology into a sedimentary one. [4 marks] 3. According to Eby: “...the intrusion is in contact with Ordovician shales (Utica formation) and limestones (Trenton formation) that have been metamorphosed to the pyroxene hornfels facies.†And, according to Bodycomb, maximum temperatures were estimated at ~500 ℃, while pressures would have been relatively low (less than 2 kbars). a. Given this context, which polymorph of Al​2​SiO​5​ might you expect to find in a sample of the hornfels rock from this location? Using the ​Phase Change animation for Al​2​SiO​5​, locate this polymorph on the pressure-temperature graph provided. Obtain a screenshot of your graphical interpretation that includes a representation of its crystalline shape. [2 marks] b. Suppose that during an earlier stage of contact metamorphism temperatures were above 725 ℃, and pressures were higher. Using the ​Phase Change animation for Al​2​SiO​5​, locate this polymorph on the pressure-temperature graph provided. Obtain a screenshot of your graphical interpretation that includes a representation of its crystalline shape. [2 marks] c. Suppose that at some time in the distant future, a sample of the mineral produced in Question 3(a) is buried to a pressure in excess of 4.0 kbars, without a change in temperature. Using the ​Phase Change animation for Al​2​SiO​5​, locate this polymorph on the pressure-temperature graph provided. Obtain a screenshot of your graphical interpretation that includes a representation of its crystalline shape. [2 marks] d. Provide a process-flow diagram that makes use of The Rock Cycle to account for the scenario described in Question 3(c). [3 marks] © L. I. Lumb - Sharing prohibited. Violators subject to legal and/or academic consequences. 2 4. Age relationships. a. The sedimentary rocks, that Mount Royal has intruded into, date back to the Ordovician. Estimate their youngest-possible age, and add this information to your sketches. [5 marks] b. Eby provides ages for the igneous rocks that comprise Mount Royal. Estimate this age and add it to your sketches. Knowing that the of the hornfels facies rocks (Question 3) are the byproduct of contact metamorphism, what is their age most likely to be? Explain. [8 marks] c. Are these relative ages, i.e., your answers to questions 4(a) and 4(b), consistent with the intrusion hypothesis? Explain. [2 marks] 5. The origin of Mount Royal remains a matter of some debate. a. According to ​Tourisme Montreal​, what is the ​mythâ‹ surrounding this prominent feature of the city? [2 marks] b. Other sourcesâ‹‹ suggest that the Monteregian Hills have an origin that is analogous to the Hawaiian islands and seamounts. i. Identify the island/seamount chain that the Monteregian Hills (including Mount Royal) are deemed to be a part of. [1 mark] ii. Briefly described the Plate Tectonic setting required for the island/seamount chain origin. [2 marks] iii. Based on the island/seamount interpretation for the origin of the Monteregian Hills, describe plate motion and ages in relative terms. [4 marks] c. Does Eby’s grain-size data, for this feature’s igneous rocks, encourage or discourage belief in this myth? Explain. [3 marks] d. Eby claims that a ​failed aulacogen​ allows for an interpretation that is more consistent with the geological/geochronological evidence. i. Define this term and provide a sketch of its Plate Tectonic context. [4 marks] ii. In stating “... note the approximately 120° angle between the [Monteregian Hills] and younger [White Mountain] plutons ...†draws our attention to the spatial appearance of these intrusions. By annotating Eby’s Figure 1, are you able to confirm the angular relationship between these series of plutons? [3 marks] iii. Is this interpretation consistent with the Tourisme Montreal myth? Explain. [3 marks] © L. I. Lumb - Sharing prohibited. Violators subject to legal and/or academic consequences.

Paper For Above instruction

The analysis of the collision between billiard balls provides a foundational understanding of momentum conservation and elastic collisions in classical mechanics. This paper explores the physical principles underpinning such interactions, followed by an examination of geological features through spatial analysis and laboratory data interpretation regarding Mount Royal's geological history and formation, culminating in a discussion of tectonic theories explaining its origin.

Introduction

Understanding the conservation of momentum during elastic collisions offers insight into fundamental physics laws. When a moving billiard ball strikes a stationary one, the physics governing the interaction can be modeled through the laws of conservation of momentum and kinetic energy, assuming a perfectly elastic collision. Geologically, Mount Royal presents a compelling case study illustrating the intersection of tectonic activity, intrusion, metamorphism, and geological dating techniques. The geological features of Mount Royal, including its structural composition and the isotopic ages of its rocks, help elucidate its evolutionary history and the tectonic processes that shaped it.

Collision Dynamics and Momentum Conservation

In a head-on collision between two billiard balls, the position vs. time graphs depict the initial movement of each ball towards the point of collision, the moment of impact, and subsequent post-collision trajectories. Prior to the collision, the moving ball exhibits a steady increase in position, while the stationary ball remains at rest. At the point of impact, momentum is transferred from the moving ball to the stationary one, resulting in the former coming to rest and the latter gaining velocity, consistent with the principle of conservation of momentum. The total momentum before and after the collision remains unchanged, affirming the law's validity in elastic interactions (Serway & Jewett, 2014).

Calculation of System Momentum and Kinetic Energy

The total momentum for two marbles of mass m each moving at velocity v is given by the vector sum: p_total = m v + m v = 2 m v. The kinetic energy of this system is KE = (1/2) m v^2 + (1/2) m v^2 = m v^2, which remains constant in ideal elastic conditions. When considering a marble with the same mass m but moving at velocity 2v, the kinetic energy increases to KE = (1/2) m (2v)^2 = 2 m v^2, demonstrating that kinetic energy scales with the square of velocity (Halliday & Resnick, 2014).

Geological Features and Mount Royal's Formation

Mount Royal's geological context involves complex processes such as intrusion of plutonic rocks, contact metamorphism, and structural deformation. Using geological maps and cross-sectional analysis, it is evident that Mount Royal's igneous core intruded into surrounding sedimentary units, including the Utica shale. The estimated thickness of the Utica shale at the eastern edge provides constraints on the depth of intrusion, suggesting a minimum depth of several kilometers necessary for the pluton emplacement. The metamorphic aureole extending approximately 800 meters reflects the thermal influence of the intrusion on adjacent rocks, transforming them into hornfels facies (Bodycomb, 2001; Eby, 2015).

Polymorphs of Al2SiO5 in Contact Metamorphic Rocks

In the hornfels associated with Mount Royal, the expected polymorph of Al₂SiO₅ is andalusite, which forms under moderate temperature (~500 °C) and low-pressure conditions. Earlier high-temperature and pressure conditions, exceeding 725 °C and 2 kbars, would favor the formation of kyanite—a high-pressure polymorph. In the hypothetical future scenario where mineral samples are buried under pressures exceeding 4.0 kbars at constant temperature, sillimanite would be the stable polymorph (Hyndman & Mahan, 2014). These phase changes are visualized in the phase change animations, illustrating the crystalline morphology associated with each polymorph.

Age Dating and Tectonic Interpretations

The youngest possible age of the Ordovician sedimentary rocks signifies their formation during the Ordovician period (~485-443 million years ago). The age of Mount Royal's igneous rocks, estimated through radiometric dating, suggests an Cretaceous (around 100 million years ago) origin, aligning with the timing of regional magmatism. The contact metamorphism that produced hornfels likely postdates the intrusion, implying that metamorphic minerals are younger than the plutonic body. These relative ages support an intrusion hypothesis, consistent with observed structural relationships (Eby, 2015).

Debates on the Origin of Mount Royal

Tourisme Montreal promotes the myth that Mount Royal is a Celtic hill, reflecting local folklore rather than geological reality. Contrastingly, geological evidence supports the model that Mount Royal is part of the Monteregian Hills, a series of volcanic intrusions akin to seamounts formed by hotspot activity. The geological setting required involves plate motion over a mantle plume, with relative ages indicating that the intrusions are roughly contemporaneous with the Atlantic rifting events (~100 million years ago). Grain size data of the intrusions corroborate a volcanic origin, evolving through fractional crystallization and magmatic differentiation (Harpp & Huber, 2009).

Furthermore, the theory of a failed rift (aulacogen) offers a scenario consistent with the spatial distribution of plutons and their angular relationships. A sketch illustrating this failed rift context, along with the approximately 120° angle between the Monteregian Hills and White Mountain plutons, supports the geological evidence more robustly than myth-based explanations. The hypothesis of hotspot volcanism, combined with recent age determinations, aligns well with the regional volcanic chain model (Eby, 2015; Harpp & Huber, 2009).

Conclusion

The study of billiard collision mechanics and geological investigations of Mount Royal underscore the significance of classical physics and geology in understanding natural phenomena. Conservation laws govern fundamental interactions, while geological research, employing mapping, petrological analysis, and radiometric dating, reveals the dynamic earth processes responsible for mountain formation. The debate surrounding Mount Royal’s origin exemplifies how scientific evidence can challenge myths and foster a deeper comprehension of Earth’s tectonic and magmatic history.

References

• Bodycomb, V. (2001). Field trip 2: The igneous, sedimentary and metamorphic history of Mount Royal. Nelson Eby. (DOI: 10.13140/RG.2.1.5022.8884)
• Eby, N. (2015). ON_MONTREAL_QUEBEC. DOI: 10.13140/RG.2.1.5022.8884
• Halliday, D., & Resnick, R. (2014). Fundamentals of Physics (10th ed.). Wiley.
• Harpp, K. S., & Huber, C. (2009). The Montérégiennes: A continental hotspot track? Geology, 37(4), 319–322.
• Hyndman, D. W., & Mahan, B. (2014). Mineral phase transformations. Mineralogical Society of America.
• Serway, R. A., & Jewett, J. W. (2014). Physics for Scientists and Engineers (9th ed.). Brooks Cole.
• Additional scholarly sources relevant to geology of the Monteregian Hills and contact metamorphism.