Problem 469 - JEE Physics

Question

If the K.E. of a body is increased by \(44 \%\), its momentum will increase by....... (A) \(20 \%\) (B) \(22 \%\) (C) \(2 \%\) (D) \(120 \%\)

Step-by-Step Solution

Answer

The momentum will increase by \(20\%\).

Step 1: Express kinetic energy in terms of momentum

We begin by writing the equation for the kinetic energy (K.E.) of a body. \[K.E. = \frac{1}{2}mv^2\] Next, we recall the equation for the momentum (p) of a body, which is given by: \[p = mv\] Now, we express the kinetic energy in terms of momentum. To do this, we square the momentum equation to obtain: \[p^2 = m^2v^2\] We then solve this equation for \(v^2\): \[v^2= \frac{p^2}{m^2}\] Now, substitute the expression for \(v^2\) in the kinetic energy equation: \[K.E. = \frac{1}{2m}\cdot\frac{p^2}{m}\]

Step 2: Determine the percentage change in momentum

Let the initial kinetic energy be \(K.E._1\) and the final kinetic energy be \(K.E._2\). Similarly, the initial momentum is \(p_1\) and the final momentum is \(p_2\). We are given that the kinetic energy increases by 44%: \[K.E._2 = 1.44\cdot K.E._1\] Substitute the expressions for kinetic energy in terms of momentum, we get: \[\frac{1}{2m}\cdot\frac{p_2^2}{m} = 1.44 \cdot \frac{1}{2m}\cdot\frac{p_1^2}{m}\] Simplify: \[\frac{p_2^2}{p_1^2} = 1.44\] Now, we take the square root of both sides of the equation, and we get: \[\frac{p_2}{p_1} = 1.2\]

Step 3: Calculate the percentage increase in momentum

Now that we found the ratio of the final to the initial momentum, we can calculate the percentage increase in momentum. The percentage increase is given by the formula: \[\% \text{ increase} = \frac{p_2 - p_1}{p_1} \times 100\%\] Using the ratio we found in the previous step: \[\% \text{ increase} = \frac{1.2p_1 - p_1}{p_1} \times 100\%\] Simplify the equation: \[\% \text{ increase} = 0.2 \times 100\% = 20\%\] So the correct answer is: (A) \(20\%\)

Key Concepts

Momentum

Momentum is a fundamental concept in physics, describing the quantity of motion an object possesses. It is defined as the product of an object's mass and velocity, expressed mathematically as:

• \( p = mv \)

This means momentum depends directly on how heavy an object is (mass) and how fast it is moving (velocity).

Momentum is a vector quantity, meaning it has both magnitude and direction. This attribute makes it crucial in analyzing collisions and calculating resultant forces in mechanics.

In real-world contexts, momentum helps us understand various phenomena, such as why it takes more effort to stop a fast-moving vehicle compared to a slow-moving bicycle.

Understanding the concept of momentum not only aids in solving physics problems related to motion and collisions but also in grasping how changes in velocity affect an object's overall motion.

Kinetic Energy

Kinetic energy refers to the energy that an object possesses due to its motion. It is mathematically expressed as:

• \( K.E. = \frac{1}{2} mv^2 \)

Here, \( m \) is the mass of the object, and \( v \) is its velocity. This equation highlights that kinetic energy varies with the square of the object's velocity, making speed a critical factor.

The relationship between kinetic energy and momentum arises from these equations. Since momentum \( p \) is \( mv \), the kinetic energy can also be expressed in terms of momentum:

• \( K.E. = \frac{p^2}{2m} \)

This expression helps understand how energy changes with changes in motion. If kinetic energy increases, momentum will increase as well, though not in a simple one-to-one ratio due to the quadratic nature of the velocity term in kinetic energy.

Kinetic energy is crucial in various applications, from calculating the speed required to launch rockets to understanding the energy conducted via mechanical systems.

Percentage Change Calculation

Percentage change calculations are invaluable when determining how one measurement differs from another in relative terms. This is often used to describe changes in physical quantities, such as momentum and kinetic energy in our exercise.

To calculate percentage change, you use the formula:

• \( \% \text{ Increase} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 \% \)

This formula gives you the amount by which a quantity has increased or decreased, expressed as a percentage of the original value.

For instance, in the context of the exercise, after determining the ratio \( \frac{p_2}{p_1} = 1.2 \), indicating a 20% increase in momentum, the percentage increase formula allows one to conclude accurately. By multiplying the fractional increase \( 0.2 \) by 100, we determine that the momentum increases by 20%, helping us solve the problem effectively.

This method is widely applicable in various fields, such as finance, physics, and statistics, to quantify changes and facilitate comparison in a straightforward manner.