Problem 33 - Master Resource Book in JEE Main Physics

Question

If the kinetic energy of a body is directly proportional to time \(t\), the magnitude of the force acting on the body is (a) directly proportional to \(\sqrt{t}\) (b) inversely proportional to \(\sqrt{t}\) (c) directly proportional to the speed of the body (d) inversely proportional to the speed of the body

Step-by-Step Solution

Answer

The force is inversely proportional to \( \sqrt{t} \) (option b).

Step 1: Understanding Kinetic Energy Proportionality

Given that kinetic energy (KE) is directly proportional to time, we write \( KE = k \cdot t \) where \( k \) is a proportionality constant. Kinetic energy is also given by the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is mass and \( v \) is velocity.

Step 2: Equating Expressions for Kinetic Energy

Set the expressions for kinetic energy equal to each other: \( \frac{1}{2}mv^2 = k \cdot t \). Rearranging gives \( v^2 = \frac{2k}{m}t \). This implies that \( v = \sqrt{\frac{2k}{m}} \sqrt{t} \).

Step 3: Finding the Expression for Force

Considering force \( F = m \frac{dv}{dt} \), calculate \( \frac{dv}{dt} \). From \( v = c \sqrt{t} \), where \( c = \sqrt{\frac{2k}{m}} \), differentiate: \( \frac{dv}{dt} = \frac{c}{2\sqrt{t}} \). Thus, \( F = m \cdot \frac{c}{2\sqrt{t}} = \frac{mc}{2} \cdot \frac{1}{\sqrt{t}} \).

Step 4: Analyzing Force Proportionality

From the expression for \( F \), \( F \propto \frac{1}{\sqrt{t}} \). Therefore, the magnitude of the force is inversely proportional to \( \sqrt{t} \). This means option (b) is correct: the force is inversely proportional to \( \sqrt{t} \).

Key Concepts

Force Proportionality

In physics, the concept of force proportionality is about examining how force changes in relation to another variable. For this exercise, we need to understand how the force exerted on an object relates to time. The original problem states that the kinetic energy (KE) is directly proportional to time, meaning if time increases, the kinetic energy also increases proportionally. To find how force relates to time, we need to use the expression for kinetic energy: \( KE = \frac{1}{2}mv^2 \). By equating this with \( KE = k \cdot t \), we relate velocity with time, which helps in determining the force. After deriving, we find that force, given as \( F = m \frac{dv}{dt} \), is inversely proportional to the square root of time, \( F \propto \frac{1}{\sqrt{t}} \). Hence, as time increases, the force on the body decreases.

Velocity and Time Relationship

Understanding the velocity and time relationship is crucial in solving problems related to motion. According to the provided solution, since kinetic energy \( KE \) changes with time \( t \), this also affects how the velocity \( v \) of the body changes.From the step-by-step solution, we saw that \( KE = \frac{1}{2}mv^2 = k \cdot t \). Solving for velocity, we derive \( v = \sqrt{\frac{2k}{m}} \sqrt{t} \), showing us that velocity is directly proportional to the square root of time. Hence, as time increases, the velocity of the body increases with the square root. This relationship helps us determine how changes in time affect the speed of an object.

Differentiation in Physics

Differentiation plays a pivotal role in physics, especially in understanding how quantities like velocity and force change over time. In this problem, differentiation is required to find the rate of change of velocity with respect to time, \( \frac{dv}{dt} \), which is essential to determine force.Given \( v = c \sqrt{t} \), where \( c \) is a constant, differentiating with respect to time gives \( \frac{dv}{dt} = \frac{c}{2\sqrt{t}} \). This rate tells us how velocity changes as time progresses. Multiplying this by the mass \( m \) gives us the force: \( F = m \cdot \frac{dv}{dt} = \frac{mc}{2\sqrt{t}} \). This shows not only the application of differentiation but also its importance in deriving formulas that relate physical quantities.