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Chapter 8: Problem 12

(a) The coefficient of restitution between two colliding objects is less than } 1 \text {. }

Short Answer

Expert verified
The collision is not perfectly elastic; some kinetic energy is lost.

Step by step solution

01

- Understand Coefficient of Restitution

The coefficient of restitution (e) is a measure of how elastic a collision is between two objects. It is defined as the ratio of the relative speed after collision to the relative speed before collision along the line of collision. Mathematically, it's given by: e = \frac{v_2 - v_1}{u_1 - u_2}where:- \( v_1 \) and \( v_2 \) are the velocities of the two objects after collision,- \( u_1 \) and \( u_2 \) are their velocities before collision.
02

- Analyze the Given Condition

The problem states that the coefficient of restitution is less than 1, i.e., e < 1.This implies that some kinetic energy is lost during the collision. A perfectly elastic collision would have a coefficient of restitution equal to 1 (no kinetic energy is lost), while a completely inelastic collision would have a coefficient of restitution equal to 0 (maximum kinetic energy is lost).
03

- Interpret the Physical Meaning

When the coefficient of restitution is less than 1, it means that the colliding objects do not bounce back to their original velocities. Instead, there is a loss of kinetic energy, which could be transformed into other forms of energy such as heat, sound, or deformation. Thus, the final velocities of the objects after the collision will be lower than they were before the collision.

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Key Concepts

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

Elastic Collisions
In physics, an elastic collision is a collision where there is no net loss in kinetic energy in the system as a result of the collision. During an elastic collision, objects bounce off each other without any deformation or generation of heat.
Elastic collisions are characterized by the coefficient of restitution (e) being equal to 1. This means that the total kinetic energy and the momentum of the system are conserved.
For example, when two billiard balls collide, they typically undergo an elastic collision, as they bounce back without losing kinetic energy.
Kinetic Energy Loss
Kinetic energy loss occurs in collisions where the coefficient of restitution (e) is less than 1. This indicates that the collision is not perfectly elastic and some energy is transformed into other forms.
When two objects collide and e < 1, part of the initial kinetic energy is changed into heat, sound, or deformation of the colliding bodies. The amount of kinetic energy retained is lower after the collision.
For instance, consider a car crash. The collision causes the cars to deform and generate heat and sound, leading to a considerable loss of the system's initial kinetic energy.
Velocity Change
In any collision, the velocities of the colliding objects will change due to the impact. The change in velocity depends on the coefficient of restitution between the objects.
The formula for e, given by \(e = \frac{v_2 - v_1}{u_1 - u_2}\), helps us determine how much the velocities change.
If e = 1, the velocities change, but the kinetic energy remains constant. If e = 0, the objects stick together after collision, showing maximum kinetic energy loss.
The change in velocity also indicates the direction and speed of the objects post-collision, which is crucial for understanding the outcomes of real-world impacts, like those in sports or vehicle collisions.

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Most popular questions from this chapter

A smooth sphere \(\mathrm{A}\) of mass \(2 m\), moving on a horizontal plane with speed \(u\), collides directly with another smooth sphere B of equal radius and of mass \(m\), which is at rest. If the coefficient of restitution between the spheres is \(e\), find their speeds after impact. The sphere B later rebounds from a perfectly elastic vertical wall, and then collides directly with A. Prove that after this collision the speed of B is \(9(1+e)^{2} u\) and find the speed of \(\mathrm{A}\).

Two masses of 20 and 10 units. moving in the same direction at speeds of 16 and 12 units respectively collide and stick together. Find the velocity of the combined mass immediately afterwards.

A smooth plane is fixed at an inclination \(30^{\circ}\) with its lower edge at a height \(a\) above a horizontal table. Two particles \(\mathrm{P}\) and \(\mathrm{Q}\), each of mass \(m\), are connected by a light inextensible string of length \(2 a\), and \(\mathrm{P}\) is held at the lower edge of the inclined plane while \(\mathbf{Q}\) rests on the table vertically below \(\mathrm{P}\). The particle \(\mathrm{P}\) is then projected with velocity \(u(u>\sqrt{g a}\) ) upwards along a line of greatest slope of the plane. Find the impulsive tension in the string when \(\mathrm{Q}\) is jerked into motion. Determine the magnitude of \(u\) if \(\mathrm{Q}\) just reaches the lower edge of the plane, and the tension in the string while \(Q\) is moving.

Water issues from a pipe. whose cross section is \(c \mathrm{~m}^{2}\), in a horizontal jet with velocity \(v \mathrm{~ms}^{-1}\). What force must be exerted by a shield placed perpendicular to the jet to bring the water to a horizontal stop? (The mass of \(1 \mathrm{~m}^{3}\) of water is \(10^{3} \mathrm{~kg}\) ).

Two particles \(\mathrm{A}\) and \(\mathrm{B}\) collide directily head-on and bounce. Find their speeds immediately after impact. (a) The mass of \(\mathrm{A}\) is twice the mass of B. (b) Just before impact the speed of \(\mathrm{A}\) is \(4 \mathrm{~ms}^{-1}\) and that of \(\mathrm{B}\) is \(3 \mathrm{~ms}^{-1} .\) (c) No kinetic energy is lost by the impact.
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