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Chapter 6: Problem 31

A smooth sphere of mass \(m\) is moving on a horizontal plane with a velocity \(3 \hat{i}+\hat{j}\), It collides with a vertical wall which is parallel to the vector \(\hat{j}\). If the coefficient of restitution between the sphere and wall is \(\frac{1}{2}\), Then (a) The velocity of sphere after impact \(-\frac{3}{2} \hat{i} \mid \hat{j}\) (b) The loss in kinetic energy causes by the impact \(-\frac{27}{8} J\) (c) The impulse \(\vec{J}\) that act on the sphere is \(\frac{9}{2} m \hat{i}\) (d) None

Short Answer

Expert verified
Options (a), (b), and (c) are correct.

Step by step solution

01

Identify Components of Initial Velocity

The initial velocity of the sphere is given as \(3\hat{i} + \hat{j}\). This means the velocity component perpendicular to the wall (parallel to the \(\hat{i}\) direction) is \(3\hat{i}\), and the component parallel to the wall (which does not change) is \(\hat{j}\).
02

Apply Coefficient of Restitution

The coefficient of restitution, \(e\), is given as \(\frac{1}{2}\). The formula for the coefficient of restitution is \(e = \frac{v_f}{v_i}\), where \(v_f\) and \(v_i\) are the final and initial velocities in the normal direction. Thus, \(v_f = e \times v_i = \frac{1}{2} \times 3 = \frac{3}{2}\). The direction is reversed, so the final velocity component in the \(\hat{i}\) direction is \(-\frac{3}{2}\hat{i}\).
03

Determine Final Velocity

The velocity of the sphere after the impact in vector form is \(-\frac{3}{2}\hat{i} + \hat{j}\). This is because the \(\hat{j}\) component remains unchanged due to its parallelism with the wall.
04

Calculate Change in Kinetic Energy

The initial kinetic energy is \(KE_{initial} = \frac{1}{2} m (3^2 + 1^2) = \frac{1}{2} m (9 + 1) = 5m\). The final kinetic energy is \(KE_{final} = \frac{1}{2} m ((-\frac{3}{2})^2 + 1^2) = \frac{1}{2} m (\frac{9}{4} + 1) = \frac{13}{8}m\). The loss in kinetic energy is \(5m - \frac{13}{8}m = \frac{27}{8}m\).
05

Calculate Impulse

Impulse, \(\vec{J}\), is the change in momentum. Since the wall is parallel to \(\hat{j}\), the change in momentum is only in the \(\hat{i}\) direction: \(J = m(v_f - v_i) = m\left(-\frac{3}{2} - 3\right) = -\frac{9}{2}m\). The magnitude of the impulse acting on the sphere is \(\frac{9}{2}m\hat{i}\).
06

Final Step: Verify Options

Now we compare the calculated values with the provided answers. (a) matches with our calculated velocity; (b) matches with our calculated loss of kinetic energy; (c) calculated impulse matches the option given. Thus, all provided answers (a), (b), and (c) are correct.

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Key Concepts

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

Coefficient of Restitution
The coefficient of restitution is a measure of how bouncy a collision is. It's often denoted by the symbol \(e\) and ranges from 0 to 1. A value of 1 indicates a perfectly elastic collision, where there's no loss of kinetic energy. A value of 0 represents a perfectly inelastic collision where objects stick together, losing maximum kinetic energy.

In our exercise, the coefficient of restitution is given as \(\frac{1}{2}\). This means the collision between the sphere and the wall is partially elastic.
The formula for the coefficient of restitution is:
  • \(e = \frac{v_f}{v_i}\)
where:
  • \(v_f\) is the final velocity in the direction perpendicular to the wall.
  • \(v_i\) is the initial velocity in the direction perpendicular to the wall.
This helps us understand how much of the sphere's speed is retained after the bounce.
Kinetic Energy
Kinetic energy represents the energy an object possesses due to its motion. It's a crucial quantity in collision physics because it can change during collisions. The formula for kinetic energy (\(KE\)) is:
  • \(KE = \frac{1}{2}mv^2\)
where:
  • \(m\) is the mass of the sphere.
  • \(v\) is the velocity magnitude.
In the given exercise, we observed that the initial and final kinetic energies differ, indicating that energy is lost in the collision. The loss in kinetic energy is calculated as the difference between the initial and final kinetic energies. It tells us how much energy was dissipated, possibly as sound or heat, due to the collision.
Impulse
Impulse is the effect of an average force acting over a period of time, resulting in a change in momentum for an object. It's a vector quantity, which means it has both magnitude and direction. The formula for impulse (\(\vec{J}\)) is:
  • \(\vec{J} = \Delta \vec{p}\)
  • Where \(\Delta \vec{p}\) is the change in momentum.
The direction of the impulse aligns with the direction of the force applied. In the case of the smooth sphere colliding with a vertical wall, impulse causes a change in the component of momentum perpendicular to the wall. In mathematical terms:
  • \(\vec{J} = m(v_f - v_i)\)
This calculation shows how much force was needed to bring about the observed change in the sphere's motion after hitting the wall.
Momentum
Momentum is a key concept when considering the motion of objects, especially during collisions. It's defined as the product of an object's mass and velocity:
  • \(\vec{p} = m\cdot\vec{v}\)
where:
  • \(m\) represents the mass.
  • \(\vec{v}\) is the velocity vector.
Momentum is conserved in isolated systems, meaning the total momentum before a collision equals the total momentum after, provided no external forces act on the system.

In this exercise, since the wall is non-movable, only the sphere's momentum in the direction perpendicular to the wall (\(\hat{i}\) direction) changes. Understanding how momentum changes gives insight into the forces at play during the collision.
Velocity Components
Understanding the components of velocity is crucial, especially in multi-dimensional motion. Velocity, a vector quantity, has both magnitude and direction.

In our problem, the sphere's velocity is initially expressed as \(3\hat{i} + \hat{j}\). This means:
  • \(3\hat{i}\) is the component of velocity perpendicular to the wall.
  • \(\hat{j}\) is the component of velocity parallel to the wall.
During the collision, only the perpendicular component (\(3\hat{i}\)) changes according to the coefficient of restitution. The parallel component (\(\hat{j}\)) remains constant as the wall exerts no force in this direction. This concept is pivotal in breaking down complex motions into simpler parts for analysis, showing how different forces affect separate directional components.

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Most popular questions from this chapter

A block of mass \(2.0 \mathrm{~kg}\) moving at \(2.0 \mathrm{~m} / \mathrm{s}\) collides head on with another block of equal mass kept at rest. If the actual loss in kinetic energy is half of the maximum loss in kinetic energy, find the coefficient of restitution. (a) 2 (b) \(\frac{1}{2}\) (c) \(\sqrt{2}\) (d) \(\frac{1}{\sqrt{2}}\)

Statement-1: Internal forces cannot change kinetic energy of a system of particles. Statement-2 : Internal forces cannot change mmentum of a system of particles.

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Mass is non uniformly distributed over the rod of length \(l\). Its linear mass density varies linearly with length as \(\lambda=k x\). The position of centre of mass is given by: (a) \(\frac{2 l}{5}\) (b) \(\frac{1}{3}\) (c) \(\frac{3 l}{4}\) (d) \(\frac{2 l}{3}\)
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