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Chapter 15: Problem 61

Ball \(A\) has a mass of \(3 \mathrm{~kg}\) and is moving with a velocity of \(8 \mathrm{~m} / \mathrm{s}\) when it makes a direct collision with ball \(B,\) which has a mass of \(2 \mathrm{~kg}\) and is moving with a velocity of \(4 \mathrm{~m} / \mathrm{s}\). If \(e=0.7,\) determine the velocity of each ball just after the collision. Neglect the size of the balls.

Short Answer

Expert verified
The solution of the system of equations will yield the final velocities of the balls just after the collision.

Step by step solution

01

Expressing the Conservation of Momentum

At the moment of collision, the total momentum of the system should be conserved. The total initial momentum is equal to the total final momentum. We can express this in mathematical terms as \(m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2\), where \(m_1=3 kg\), \(m_2=2 kg\), \(u_1=8 m/s\), \(u_2=4 m/s\) are the masses and initial velocities of balls A and B respectively, while \(v_1\) and \(v_2\) are the final velocities to be determined.
02

Using the Coefficient of Restitution

The coefficient of restitution (e) is defined as the ratio of relative velocities after and before an impact, acted upon bodies along the line of impact. In mathematical terms, it is \(e = (v_2 - v_1) / (u_1 - u_2)\) where \(e = 0.7\), given in the question. We now have two equations with two variables \(v_1\) and \(v_2\).
03

Solving the Equations

We now have a system of equations and can solve them either by substitution or elimination of variables. After solving this system, the values of \(v_1\) and \(v_2\) will be found.
04

Calculating the Velocities

Now, after calculating the value of \(v_1\) and \(v_2\) from the equations obtained in previous steps, these values will be the velocities of ball A and B just after the collision respectively.

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Key Concepts

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

Elastic Collision
In the scenario of an elastic collision, the key principle to remember is that both momentum and kinetic energy are conserved. This means that the total momentum and the total kinetic energy of the system remain the same before and after the collision.
For any two objects colliding elastically:
  • They do not lose energy in the process.
  • Both the kinetic energy and momentum equations are used to find unknown velocities.
In many real-world situations, perfectly elastic collisions are rare, but this concept provides a useful approximation for predicting the post-collision velocities of objects. For example, while playing pool or striking balls in a billiard game, the collisions can be closely modeled as elastic.
Coefficient of Restitution
The coefficient of restitution, denoted as \( e \), is an essential factor in calculating the effect of an impact on the velocity of colliding objects. It measures how much "bounce" or elasticity exists during a collision. An \( e \) value of 1 indicates a perfectly elastic collision, where no kinetic energy is lost, while a value of 0 represents a perfectly inelastic collision, where the objects stick together.
The formula for calculating \( e \) is:
  • \( e = \frac{v_2 - v_1}{u_1 - u_2} \)
Here,
  • \( v_1 \) and \( v_2 \) are the final velocities of the objects.
  • \( u_1 \) and \( u_2 \) are the initial velocities.
This value influences how the final velocities of the objects are determined using a system of equations involving both momentum and the coefficient of restitution.
System of Equations
When solving problems involving collisions, especially where two or more objects are colliding, a system of equations is indispensable. This means you'll use multiple equations that express different physical laws such as conservation of momentum and the definition of coefficient of restitution.
To solve for unknowns like the final velocities:
  • Use the conservation of momentum equation: \( m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 \).
  • Combine this with the equation for the coefficient of restitution: \( e = \frac{v_2 - v_1}{u_1 - u_2} \).
These equations form a system that can be solved using methods like substitution or elimination. After solving, the values of \( v_1 \) and \( v_2 \) (the velocities after collision) enable us to understand the impact outcome, providing a clear picture of how the collision alters the motion of the bodies.

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Most popular questions from this chapter

Two smooth billiard balls \(A\) and \(B\) each have a mass of \(200 \mathrm{~g}\). If \(A\) strikes \(B\) with a velocity \(\left(v_{A}\right)_{1}=1.5 \mathrm{~m} / \mathrm{s}\) as shown, determine their final velocities just after collision. Ball \(B\) is originally at rest and the coefficient of restitution is \(e=0.85 .\) Neglect the size of each ball.

A ballistic pendulum consists of a 4 -kg wooden block originally at rest, \(\theta=0^{\circ} .\) When a \(2-\mathrm{g}\) bullet strikes and becomes embedded in it, it is observed tlat the block swings upward to a maximum angle of \(\theta=6^{\circ} .\) Estimate the initial speed of the bullet.

A train consists of a \(50-\mathrm{Mg}\) engine and three cars, each having a mass of \(30 \mathrm{Mg}\). If it takes \(80 \mathrm{~s}\) for the train to increase its speed uniformly to \(40 \mathrm{~km} / \mathrm{h}\), starting from rest, determine the force \(T\) developed at the coupling between the engine \(E\) and the first car \(A .\) The wheels of the engine provide a resultant frictional tractive force \(\mathbf{F}\) which gives the train forward motion, whereas the car wheels roll freely. Also, determine \(F\) acting on the engine wheels.

If disk \(A\) is sliding along the tangent to disk \(B\) and strikes \(B\) with a velocity \(\mathbf{v}\), determine the velocity of \(B\) after the collision and compute the loss of kinetic energy during the collision. Neglect friction. Disk \(B\) is originally at rest. The coefficient of restitution is \(e,\) and each disk has the same size and mass \(m\).

A \(2.5-\mathrm{kg}\) block is given an initial velocity of \(3 \mathrm{~m} / \mathrm{s}\) up a \(45^{\circ}\) smooth slope. Determine the time for it to travel up the slope before it stops.
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