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Chapter 8: Problem 82

A body whose mass is \(m_{1}\) is acted upon at a given point \(P\) by a blow of impulse \(x\). If \(v\) and \(v^{\prime}\) be the velocities of \(P\) in the direction of \(x\) just before and just after the action of \(x\), the change in kinetic energy is: (a) \(\left(\frac{v+v^{\prime}}{2 x}\right)\) (b) \(\frac{\left(v+v^{\prime}\right) x}{2}\) (c) \(\frac{1}{4}\left(v^{\prime}+v\right) x\) (d) all of these

Short Answer

Expert verified
The correct answer is (b) \(\frac{(v+v')x}{2}\).

Step by step solution

01

Understanding Impulse

Impulse is defined as the change in momentum of an object. The formula for impulse is \( x = m_1 (v' - v) \), where \( v' \) is the final velocity and \( v \) is the initial velocity.
02

Calculate Initial Kinetic Energy

The initial kinetic energy (KE) of the object, before the impulse is applied, is given by \( KE_{initial} = \frac{1}{2} m_1 v^2 \). This is because kinetic energy is calculated using the formula \( \frac{1}{2} m v^2 \).
03

Calculate Final Kinetic Energy

After the impulse is applied, the final kinetic energy is \( KE_{final} = \frac{1}{2} m_1 v'^2 \). This uses the same kinetic energy formula, substituting the final velocity.
04

Determine the Change in Kinetic Energy

The change in kinetic energy \( \Delta KE \) is the difference between the final and initial kinetic energy: \( \Delta KE = \frac{1}{2} m_1 v'^2 - \frac{1}{2} m_1 v^2 \).
05

Simplify the Change in Kinetic Energy Expression

Simplifying \( \Delta KE \), we have:\[ \Delta KE = \frac{1}{2} m_1 (v'^2 - v^2) \]Using the identity \( a^2 - b^2 = (a + b)(a - b) \), we find:\[ \Delta KE = \frac{1}{2} m_1 (v' + v)(v' - v) \]
06

Substitute Impulse into the Expression

Replace \( v' - v \) using the impulse \( x = m_1 (v' - v) \), therefore \( v' - v = \frac{x}{m_1} \).Substituting this into the expression for \( \Delta KE \) gives:\[ \Delta KE = \frac{1}{2} m_1 (v' + v) \frac{x}{m_1} \]This simplifies to:\[ \Delta KE = \frac{1}{2} (v' + v)x \].
07

Identify the Correct Option

Looking at the options, the expression we derived \( \frac{1}{2}(v' + v)x \) matches option (b), \( \frac{(v + v')x}{2} \). Thus, the correct answer is option (b).

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Key Concepts

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

Kinetic Energy
Kinetic energy is the energy an object possesses due to its motion. It depends on two primary factors: the mass of the object and its velocity. The formula for kinetic energy is given by \[ KE = \frac{1}{2}mv^2 \] where \( m \) is the mass and \( v \) is the velocity. When a force acts on an object causing a change in its velocity, the kinetic energy of the object also changes. This is particularly important in understanding how forces like impulses can affect an object's movement. The difference between the initial and final kinetic energy of an object can tell us a lot about what happens to that energy during interactions, such as collisions or applications of force.
Momentum
Momentum is a fundamental concept in mechanics describing the motion of an object. It is the product of the mass and velocity of an object: \[ p = mv \]where \( p \) denotes momentum. Momentum is a vector quantity, which means it has both a direction and a magnitude.
  • In a collision or when an external force acts, the momentum of an object may change.
  • According to the law of conservation of momentum, in the absence of external forces, the total momentum of a system remains constant.
Impulse can be thought of as a change in momentum. The impulse delivered to an object results in a change in its velocity and hence a change in its momentum.
Velocity Change
Velocity is the speed of an object in a specified direction. A change in velocity means that either the speed changes, the direction changes, or both. In our problem, the impulse changes the velocity from \( v \) to \( v' \). This is essential in calculating the resulting effects on momentum and energy. Important points about velocity change:
  • The change in velocity due to an impulse is the primary reason for a change in momentum.
  • Velocity change can lead to a corresponding change in kinetic energy, as seen through \( \Delta KE = \frac{1}{2} (v' + v)x \).
Understanding these changes is key to figuring out real-world situations like crashes, where impulse affects objects' speeds.
Mechanics
Mechanics is a broad field of physics that deals with the motion of objects, forces, and the effects that forces have on motion. The study of mechanics is divided into two main parts:
  • Kinematics: Describes motion without considering its causes.
  • Dynamics: Looks at the effect of forces on motion.
In our exercise, impulse falls under the category of dynamics. It is the force applied over a period of time, changing an object's momentum and kinetic energy. By connecting the dots between kinetic energy, momentum, and velocity, mechanics helps us solve problems involving impulse and other forces efficiently. This foundation is crucial for understanding more complex topics like energy transfers and mechanical systems.

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Most popular questions from this chapter

A uniform rod \(A B\) of mass \(m\) and length \(l\) is at rest on a smooth horizontal surface. An impulse \(p\) is applied to the end \(B\). The time taken by the rod to turn through a right angle is : (a) \(2 \pi \frac{m l}{p}\) (b) \(2 \pi \frac{p}{m l}\) (i) \(\frac{\pi m l}{12 p}\) (d) \(\frac{\pi p}{m !}\)

A mass \(m\) is moving with a constant velocity along a line parallel to the \(x\) -axis, away from the origin. Its angular momentum with respect to the origin: (a) is zero (b) remains constant (c) goes on increasing (d) goes on decreasing

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Mark correct option or options: (a) A uniform cube has minimum moment of inertia about an axis passing through centre and is passing through opposite corners (b) The moment of inertia of a complicated shape bodies can be determined by using inertia of table (c) Displacing of axial vectors is meaningless (d) All the above
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