Problem 48 - Master Resource Book in JEE Main Physics

Question

A running man has half the kinetic energy of that of a boy of half of his mass. The man speeds up by \(1 \mathrm{~m} / \mathrm{s}\), so as to have same kinetic energy as that of the boy. The original speed of the man will be (a) \(\sqrt{2} \mathrm{~m} / \mathrm{s}\) (b) \(\sqrt{2-1} \mathrm{~m} / \mathrm{s}\) (c) \(\frac{1}{\sqrt{2}-1} \mathrm{~m} / \mathrm{s}\) (d) \(\frac{1}{\sqrt{2}} \mathrm{~m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The original speed of the man is \(\sqrt{2} \mathrm{~m/s}\).

Step 1: Understanding the Problem

We are given two individuals: a man and a boy. The man's kinetic energy is half that of the boy's, with the boy having half the mass of the man. The man increases his speed by 1 m/s to match the kinetic energy of the boy. We need to find the original speed of the man.

Step 2: Writing the Kinetic Energy Equations

Let the mass of the boy be \(m\) and his speed be \(v\). Then his kinetic energy is \(\frac{1}{2} m v^2\). The man's mass is \(2m\) and his kinetic energy is \(\frac{1}{2} (2m) u^2 = mu^2\). We're told that \(mu^2 = \frac{1}{2} \times \frac{1}{2} m v^2\), which makes \(mu^2 = \frac{1}{4} m v^2\). Dividing both sides by \(m\), we get \(u^2 = \frac{1}{4} v^2\).

Step 3: Finding the Relationship between Velocities

From \(u^2 = \frac{1}{4} v^2\), we take the square root of both sides to find \(u = \frac{v}{2}\). This means the man's original speed is half the speed of the boy.

Step 4: Calculating the Increased Speed of the Man

When the man increases his speed by 1 m/s, his new speed becomes \(u + 1\). At this speed, his kinetic energy equals the boy's, i.e., \(\frac{1}{2} (2m) (u+1)^2 = \frac{1}{2} m v^2\). Simplifying, \((u+1)^2 = v^2\).

Step 5: Solving for the Original Speed of the Man

We have two equations: \(u = \frac{v}{2}\) and \((u+1)^2 = v^2\). Substituting \(u = \frac{v}{2}\) into \((u+1)^2 = v^2\), we get \((\frac{v}{2} + 1)^2 = v^2\). Solving this yields \(\frac{v^2}{4} + v + 1 = v^2\) which simplifies to \(v^2 - v - 4 = 0\). Solving this quadratic equation gives us \(v = 2\sqrt{2}\). Substituting back \(u = \frac{v}{2}\), we get \(u = \sqrt{2}\).

Step 6: Final Answer

Thus, the original speed of the man is \(\sqrt{2} \mathrm{~m/s}\), which corresponds to option (a).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are fundamental in algebra. They are expressed in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Solutions to quadratic equations are called roots, which can be found using the quadratic formula:

• The formula is \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \).
• The term under the square root, \( b^2 - 4ac \), is known as the discriminant.

In our exercise, we encounter a quadratic equation when determining the relationship between the speeds of the man and the boy. By translating the kinetic energy relationship and the motion conditions into a quadratic form, we are able to solve for the velocity of the boy. This solution then helps us deduce the original speed of the man. Understanding quadratic equations is crucial not only for solving mathematical problems but also for applications in physics and engineering.

The Concept of Velocity

Velocity is a vector quantity that refers to the speed of an object in a particular direction. Unlike speed, which is scalar and only measures magnitude, velocity also includes the direction of motion.

• In our problem, velocity is integral for determining kinetic energy.
• The man and the boy each have a different velocity that affects their respective kinetic energies.

The exercise uses the velocities of both individuals to establish the initial conditions of the problem. By increasing the man's velocity by 1 m/s, we see how this change impacts his kinetic energy, allowing it to match that of the boy. By exploring the relationship between velocity and kinetic energy, we gain a clear understanding of how changes in speed affect the dynamics of motion.

Understanding Mass in Motion

Mass is a measure of the amount of matter in an object, and it plays a significant role in calculating kinetic energy. Kinetic energy, which measures the energy of an object in motion, is directly proportional to both mass and the square of velocity. The formula is:

• The formula for kinetic energy is \( KE = \frac{1}{2} mv^2 \), where \( m \) is mass and \( v \) is velocity.
• In our problem, the man's mass is twice that of the boy, which influences the relationship between their kinetic energies.

Understanding the role of mass helps in setting up the equations needed to find the solution. The difference in mass between the man and the boy is a key factor in determining how their kinetic energies compare and how changes in their respective velocities affect these energies. By comprehending these concepts, we are better equipped to solve problems involving motion and energy.