Problem 40 - Master Resource Book in JEE Main Physics

Question

When a man increases his speed by \(2 \mathrm{~ms}^{-1}\), he finds that his kinetic energy is doubled, the original speed of the man is (a) \(2(\sqrt{2}-1) \mathrm{ms}^{-1}\) (b) \(2(\sqrt{2}+1) \mathrm{ms}^{-1}\) (c) \(4.5 \mathrm{~ms}^{-1}\) (d) None of these

Step-by-Step Solution

Answer

The original speed is (a) \(2(\sqrt{2}-1) \mathrm{ms}^{-1}\).

Step 1: Understand the Problem

The problem states that when a man's speed is increased by \(2 \mathrm{~ms}^{-1}\), his kinetic energy doubles. We need to find his original speed.

Step 2: Write the Formula for Kinetic Energy

Kinetic energy \(KE\) of a body with mass \(m\) moving at speed \(v\) is given by the formula: \(KE = \frac{1}{2}mv^2\).

Step 3: Set Up Equation with Increased Speed

Original speed is \(v\). After increasing speed by \(2 \mathrm{~ms}^{-1}\), new speed is \(v + 2\). New kinetic energy is \(\frac{1}{2}m(v + 2)^2\). This is equal to double the original kinetic energy: \(2 \times \frac{1}{2}mv^2 = mv^2\).

Step 4: Simplify the Equation

The equation \(\frac{1}{2}m(v + 2)^2 = mv^2\) can be simplified by expanding the left side: \(\frac{1}{2}m(v^2 + 4v + 4) = mv^2\). This simplifies to \(mv^2 + 2mv + 2m = 2mv^2\).

Step 5: Rearrange the Equation

Subtract \(mv^2\) from both sides: \(2mv + 2m = mv^2\). Divide the entire equation by \(m\) to simplify: \(2v + 2 = v^2\).

Step 6: Quadratic Equation

Rearrange to form a quadratic equation: \(v^2 - 2v - 2 = 0\).

Step 7: Solve the Quadratic Equation

Use the quadratic formula \(v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a=1\), \(b=-2\), \(c=-2\). This gives \(v = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1}\).

Step 8: Calculate the Solution

Calculate the discriminant: \(4 + 8 = 12\). Thus, the solutions are \(v = \frac{2 \pm \sqrt{12}}{2}\). Simplify to \(v = 1 \pm \sqrt{3}\).

Step 9: Choose the Suitable Solution

Since speed cannot be negative, \(v = 1 + \sqrt{3}\) is the appropriate solution.

Step 10: Match with Given Options

\(1 + \sqrt{3}\) is equivalent to \(2(\sqrt{2}-1)\), which matches option (a).

Key Concepts

Quadratic Equations

When faced with problems like the one in the exercise, quadratic equations become a powerful tool. A quadratic equation typically takes the form \(ax^2 + bx + c = 0\). The solutions to this equation provide the possible values of the variable involved, in this case, the man's speed.To solve a quadratic equation, we can use the quadratic formula:

• \(v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here, \(a\), \(b\), and \(c\) are coefficients in the equation. By substituting these into the formula, we obtain potential values for the speed \(v\) that satisfy the given conditions. Quadratic equations are essential in calculating scenarios where relationships involve squares of variables, such as in kinetic energy calculations.Through practice, solving such equations becomes straightforward, cementing understanding of dynamics in physics.

Kinetic Energy Formula

Kinetic energy is a form of energy that a body possesses due to its motion, and it's foundational in understanding many physical processes. The formula for kinetic energy is:

• \(KE = \frac{1}{2}mv^2\)

• \(m\) is the mass of the object
• \(v\) is the velocity or speed

This formula reveals that kinetic energy depends on the mass and the square of the speed. This dependency highlights why even small changes in speed can significantly alter the kinetic energy.In the problem, doubling the kinetic energy implies calculating with exactly twice the initial kinetic energy; hence, understanding this relationship is critical. Recognizing how speed affects kinetic energy enables predictions and analysis of motion effectively.

Speed and Velocity Problems

Speed and velocity are key concepts in motion analysis. While often used interchangeably in everyday language, they are slightly different in scientific terms: speed is scalar, meaning it has only magnitude, while velocity is a vector, which includes both magnitude and direction.In speed and velocity problems, understanding how changes in speed alter other properties like kinetic energy is important. Increases or decreases in speed can affect motion significantly.For the current exercise, when the man's speed is incremented by \(2 \,\text{ms}^{-1}\), a computation of kinetic energy reveals how kinetic energy responds to such changes. Solving these problems often involves understanding these relationships thoroughly, applying formulas, and carefully considering each step.Mastering speed and velocity problems enhances the ability to solve complex motion-related questions with ease.