Problem 44 - Master Resource Book in JEE Main Physics

Question

A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement \(x\) is proportional to [UP SEE 2007] (a) \(x^{2}\) (b) \(\mathrm{e}^{x}\) (c) \(x\) (d) \(\log e^{x}\)

Step-by-Step Solution

Answer

(a) \(x^{2}\)

Step 1: Understanding Retardation

The retardation of a particle means it experiences a deceleration. The statement says this retardation is proportional to its displacement. Mathematically, we can express this as \( a = -kx \) where \( a \) is the acceleration (or retardation), \( k \) is the proportionality constant, and \( x \) is the displacement.

Step 2: Relate Force to Retardation

Using Newton's second law, force \( F \) is related to acceleration by \( F = ma \), where \( m \) is the mass of the particle. Substituting the expression for acceleration, we have \( F = m(-kx) = -mkx \), indicating that the force is proportional to the displacement and acts in the opposite direction.

Step 3: Analyze Loss of Kinetic Energy

The work done by the force over a displacement \( x \) results in a change in kinetic energy. For a negative constant force, the work done is \( W = \, \int F \, dx = \, \int -mkx \, dx = -\frac{1}{2}mkx^{2} \). This work corresponds to the loss of kinetic energy.

Step 4: Conclusion on Proportionality

The integral results in the change in kinetic energy being proportional to \( x^{2} \). Therefore, the loss of kinetic energy for any displacement \( x \) is directly proportional to \( x^{2} \).

Key Concepts

Retardation

Retardation is a term used when an object in motion slows down. It is often referred to as negative acceleration because it reduces the speed of an object rather than increasing it. In this problem, the retardation of a particle is proportional to its displacement.
In mathematical terms, this means that as the displacement increases, the retardation does too. We express this relationship with the equation:

• \( a = -kx \)

Here,

• \( a \) represents the acceleration (retardation)
• \( k \) is a constant of proportionality
• \( x \) is the displacement

The negative sign indicates that the force acts in the opposite direction of the movement, causing the particle to slow down. This concept is key to understanding how forces affect motion, especially in scenarios where speed is decreasing.

Work-Energy Principle

The work-energy principle provides a direct relationship between the work done on an object and the change in its kinetic energy. According to this principle, the work done by the net force on an object results in a change in its kinetic energy:

• \( ext{Work done} = ext{Change in kinetic energy} \)

For our problem, the work done over a displacement due to retardation affects the kinetic energy.
When the force does negative work, it leads to a loss in kinetic energy. In this case, the force acting on the particle is:

• \( F = -mkx \)

Taking into account this negative force, the work done over displacement \( x \) becomes:

• \( W = rac{-1}{2}mkx^2 \)

This shows that the work done, and consequently the loss in kinetic energy, is related to the square of the displacement.

Proportionality

Proportionality is a fundamental concept that describes how one quantity changes in response to another. It is expressed through equations where one variable equals a constant times another variable.
In this exercise, we encounter proportionality in retardation and energy loss. The retardation is proportional to the displacement according to:

• \( a = -kx \)

This means as the displacement increases, the retardation similarly increases.
Similarly, the loss of kinetic energy is also proportional to the displacement squared, as demonstrated by:

• \( ext{Loss in kinetic energy} = rac{-1}{2}mkx^2 \)

Understanding these proportional relationships helps in forecasting the motion's outcome simply by observing how certain variables scale with movement. These relationships are critical in analyzing systems under such variable influences.

Displacement

Displacement refers to the change in position of an object. It is a vector quantity, meaning it has both magnitude and direction. Displacement is essential in studying motion dynamics because it quantifies how far out of place an object is compared to its starting point.
In our problem, displacement (\( x \)) isn't just a measure of distance. It is a variable that influences retardation and energy loss; these depend on how much the particle has moved from an initial position.

• The greater the displacement, the more significant the effect on retardation and energy.

Since both the force acting on the particle and the resultant kinetic energy loss are related to displacement, understanding this variable is key to predicting motion in various contexts. Displacement ties together the other physical quantities in this scenario, setting the stage for the motion analysis.

Newton's Laws of Motion

Newton's laws of motion offer the foundation for understanding motion dynamics, including concepts like force and acceleration. In this context, we specifically look at the second law, which states that force equals mass times acceleration (\( F = ma \)).
By applying Newton's second law to our problem, the force (\( F \)) necessary to cause the retardation can be derived:

• \( F = m(-kx) = -mkx \)

This equation shows that the force responsible for slowing down the particle is directly proportional to both the mass of the particle and its displacement.
Furthermore, the concepts of force, displacement, and energy discussed in the exercise are interconnected through Newton's principles. These laws aid in deducing that the force derived from the particle's retardation results in a proportional loss of kinetic energy, all originating from the basic motions established by Newton.