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Chapter 2: Problem 6

A ball strikes a wall with a velocity \(\vec{u}\) at an angle \(\theta\) with the normal to the wall surface and rebounds from it at an angle \(\beta\) with the surface. Then (1) \((\theta+\beta)<90^{\circ}\), if the wall is smooth (2) if the wall is rough, coefficient of restitution \(=\tan \beta / \cos \theta\) (3) if the wall is rough, coefficient of restitution \(<\tan \beta / \cot \theta\) (4) none of the above

Short Answer

Expert verified
None of the options fully describe the situation correctly; thus, option (4) is most appropriate.

Step by step solution

01

Identify Key Concepts

The problem involves concepts of angles of incidence and reflection, including the motion of a ball striking and rebounding from a wall. We must consider the smooth and rough wall surface conditions, and understand the concept of the coefficient of restitution.
02

Analyze Smooth Wall Scenario

In a smooth wall scenario, angles of incidence and reflection (\(\theta\) and \(\beta\) respectively) add up such that \((\theta + \beta) = 90^{\circ} \) assumed for ideal conditions like mirror reflections. Here, the problem states \((\theta + \beta) < 90^{\circ}\), indicating possible deviation as the wall is assumed frictionless and could affect real-world results.
03

Examine Rough Wall & Coefficient of Restitution

The coefficient of restitution \(e\) is defined by the relative velocity of separation to the relative velocity of approach, affecting how the ball rebounds. In a rough wall condition, given \(e = \tan \beta / \cos \theta\), this hints at a relationship between the angles and restitution.
04

Consider the Condition \(e < \tan \beta / \cot \theta\)

Comparing with the previous expression of \(e\), the given option \(e < \tan \beta/\cot \theta\) signifies different outcomes on how the bounce will behave, though mathematically less likely due to equation constraints and typical definitions involving \(tan\) and \(cot\).
05

Evaluate Each Condition

Observe that under smooth conditions none of the statements \((\theta + \beta) < 90^{\circ})\) ideally hold unless considering friction as minimal leeway influenced. For rough, evaluating expressions like \(e > \tan \beta/\cos \theta\) shows potential consistency unless option (1) remains only partially reasoning valid.
06

Conclusion Based on Evaluation

None of the options fully address both mathematical validity and physical intuition consistently with provided equations and typical problem expectations. The first two statements provided conditions mildly but not consistently amply met by logical derivation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angle of Incidence
The angle of incidence is a critical concept in understanding how objects strike surfaces. It refers to the angle at which a moving object, like a ball, hits a surface. When considering this angle, it's measured between the object's path and a line perpendicular to the surface called the "normal." In physics, the angle of incidence can determine how an object behaves after striking a surface, especially when considering energy conservation and momentum equations.
  • In a smooth surface scenario, this angle helps predict how the object will rebound.
  • The angle of incidence is equal to the angle of reflection in ideal conditions, such as on a perfectly smooth surface.

Both the angle of incidence (\(\theta\)) and surface conditions significantly impact the motion of the rebounding object.
Angle of Reflection
The angle of reflection describes how an object like a bouncing ball leaves a surface. It is crucial for predicting the direction in which the object will continue after impact. This angle is measured between the rebounding path and the normal, a line perpendicular to the surface at the point of impact.
  • Ideally, the angle of reflection (\(\beta\)) should be equal to the angle of incidence if the surface is perfectly smooth, reflecting energy perfectly.
  • In real-world scenarios, several factors, such as surface roughness, can affect the equality of these angles.

In the exercise, attention is drawn to the relationship between \(\theta\) and \(\beta\). In an idealized frictionless situation, these angles might sum to \(90^{\circ}\), representing perfect symmetry. Understanding these angles can better predict object's motion post-impact.
Smooth and Rough Surfaces
Surface texture plays a significant role in determining the behavior of an object after impact. Surfaces are categorized as either smooth or rough, each affecting motion differently.**Smooth Surfaces:** - Smooth surfaces provide minimal friction. - In ideal conditions, they allow the angle of incidence to equal the angle of reflection. - Less energy is lost to friction, often leading to predictable trajectories based on geometric symmetry.**Rough Surfaces:** - Rough surfaces introduce greater friction. - This friction alters the angles of incidence and reflection, making predictions more complex due to energy loss to heat and work against friction.- On rough surfaces, additional calculations like the coefficient of restitution (\(e\)) are necessary to understand bouncing behavior.
  • The coefficient of restitution measures how bouncy a collision is, often affected by surface type.
  • For a rough wall, \(e\) includes angular relationships like \(\tan \beta\) and \(\cos \theta\), creating complex interdependencies.

By exploring these properties, students can better understand how simple changes in surface texture can influence the motion outcomes of a ball or any object.

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Most popular questions from this chapter

A block of mass ' \(m\) ' is hanging from a massless spring of spring constant \(k\). It is in equilibrium under the influence of gravitational force. Another particle of same mass ' \(m\) ' moving upwards with velocity \(u_{0}\) hits the block and sticks to it. For the subsequent motion, choose the incorrect statements: (1) Velocity of the combined mass must be maximum at natural length of the spring. (2) Velocity of the combined mass must be maximum at the new equilibrium position. (3) Velocity of the combined mass must be maximum at the instant particle hits the block. (4) Velocity of the combined mass must be maximum at a point lying between old equilibrium position and natural length.

A bullet of mass \(0.01 \mathrm{~kg}\) and travelling at a speed of \(500 \mathrm{~ms}^{-1}\) strikes a block of mass \(2 \mathrm{~kg}\) which is suspended by a string of length \(5 \mathrm{~m}\). The centre of gravity of the block is found to raise a vertical distance of \(0.2 \mathrm{~m}\). What is the speed of the bullet after it emerges from the block? (1) \(15 \mathrm{~ms}^{-1}\) (2) \(20 \mathrm{~ms}^{-1}\) (3) \(100 \mathrm{~ms}^{-1}\) (4) \(50 \mathrm{~ms}^{-1}\)

Three balls \(A, B\) and \(C\) of masses \(2 \mathrm{~kg}, 4 \mathrm{~kg}\) and \(8 \mathrm{~kg}\), respectively, move along the same straight line and in the same direction, with velocities \(4 \mathrm{~m} / \mathrm{s}, 1 \mathrm{~m} / \mathrm{s}\) and \(3 / 4 \mathrm{~m} / \mathrm{s}\). If \(A\) collides with \(B\) and subsequently \(B\) collides with \(C\), find the velocity of ball \(A\) and ball \(B\) after collision, taking the coefficient of restitution as unity. \(\overrightarrow{\text { (1) }} \overrightarrow{\text { (B) }} \overrightarrow{\text { (C) }}\) (1) \(v_{A}=3, v_{B}=9 / 4\) (2) \(v_{A}=0, v_{B}=3\) (3) \(v_{A}=3, v_{B}=0\) (4) \(v_{A}=0, v_{B}=0\)

A particle \(A\) suffers an oblique elastic collision with a particle \(B\) that is at rest initially. If their masses are the same, then after the collision (1) their KE may be equal (2) \(A\) continues to move in the original direction while \(B\) remains at rest (3) they will move in mutually perpendicular directions (4) \(A\) comes to rest and \(B\) starts moving in the direction of the original motion of \(A\)

A glass ball collides with a smooth horizontal surface ( \(x z\) plane) with a velocity \(V=a \hat{i}-b \hat{j} .\) If the coefficient of restitution of collision be \(e\), the velocity of the ball just afie the collision will be (1) \(\sqrt{e^{2} a^{2}+b^{2}}\) at angle \(\tan ^{-1}\left(\frac{a}{e b}\right)\) to the vertical (2) \(\sqrt{a^{2}+e^{2} b^{2}}\) at angle \(\tan ^{-1}\left(\frac{a}{e b}\right)\) to the vertical (3) \(\sqrt{a^{2}+\frac{b^{2}}{e^{2}}}\) at angle \(\tan ^{-1}\left(\frac{e a}{b}\right)\) to the vertical (4) \(\sqrt{\frac{a^{2}}{e^{2}}+b^{2}}\) at angle \(\tan ^{-1}\left(\frac{a}{e b}\right)\) to the vertical
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