Problem 43 - Master Resource Book in JEE Main Physics

Question

A stone of mass \(2 \mathrm{~kg}\) is projected upward with kinetic energy of \(98 \mathrm{~J}\). The height at which the kinetic energy of the body becomes half its original value, is given by (Take \(g=10 \mathrm{~ms}^{-2}\) ) (a) \(5 \mathrm{~m}\) (b) \(2.5 \mathrm{~m}\) (c) \(1.5 \mathrm{~m}\) (d) \(0.5 \mathrm{~m}\)

Step-by-Step Solution

Answer

The height is approximately 2.5 m, so the answer is option (b).

Step 1: Identify Known Values

We have the initial kinetic energy, \(K_i = 98 \text{ J}\), the mass of the stone \(m = 2 \text{ kg}\), and the acceleration due to gravity \(g = 10 \text{ m/s}^2\).

Step 2: Understand Half Kinetic Energy

We are asked to find the height where the kinetic energy becomes half its original value. Therefore, the final kinetic energy, \(K_f = \frac{K_i}{2} = \frac{98}{2} = 49 \text{ J}\).

Step 3: Use Energy Conservation Principle

According to the conservation of mechanical energy, the sum of kinetic energy and potential energy remains constant. The potential energy change can be written as \( \Delta U = mgh \). At the height \(h\), \(K_i = K_f + \Delta U\).

Step 4: Set up the Equation

Substitute the known values in the energy conservation equation:\[98 = 49 + 2 \cdot 10 \cdot h\]

Step 5: Solve for Height

Rearrange the equation to solve for \(h\):\[98 - 49 = 20h\]\[49 = 20h\]Divide both sides by 20:\[h = \frac{49}{20} = 2.45 \text{ m} \approx 2.5 \text{ m}\]

Step 6: Select the Correct Option

Compare the computed height to the answer choices. The closest answer choice is \(\text{(b)}\, 2.5 \, \text{m}\).

Key Concepts

Conservation of Mechanical Energy

The concept of conservation of mechanical energy is fundamental in physics. It tells us that the total mechanical energy (the sum of kinetic and potential energy) in a closed system remains constant, provided that no other forces, like friction or air resistance, act on the system. For our stone, as it moves upward after being projected, its kinetic energy is converted to potential energy due to gravity.
The formula governing this conservation is straightforward:

• Initial Energy: The total energy at the start (all kinetic at point of projection) = \( K_i = 98 \, \text{J} \).
• Final Energy: The energy becomes a combination of kinetic (\( K_f = 49 \, \text{J} \) in the problem) and potential energy when the stone is at a height \( h \).
• At any point, \( K_i = K_f + \Delta U \), where \( \Delta U \) is the change in potential energy.
• This conservation helps us calculate the height where kinetic energy becomes half by equating it to potential energy changes.

This principle enables us to see how energy transfers between different forms without loss, providing a powerful tool for solving dynamic problems like projectile motion.

Potential Energy

Potential energy is the stored energy of an object due to its position or state. In our context, it's the energy stored because of the stone’s height above the ground, driven by gravitational force.
The potential energy is determined by the formula:

• \( U = mgh \)
• where \( m \) is mass, \( g \) is the gravitational constant, and \( h \) is the height.

When the stone rises, it loses kinetic energy and gains potential energy, maintaining total energy conservation. In the problem, when kinetic energy becomes half, the remaining energy is stored as potential energy:

• Initial potential energy at the ground level is zero.
• At height \( h \), potential energy \( = mgh = 49 \, \text{J} \), using the values provided.

Understand this transformation helps visualize how energy processes work during the stone's movement. It's crucial to remember that while the kinetic energy decreases, potential energy increases and vice versa, keeping the total energy the same.

Projectile Motion

Projectile motion is a form of motion experienced by an object that is projected into the air and influenced only by gravity. Our stone experiences projectile motion when thrown upwards.
It involves two components:

• A vertical component affected by gravity.
• A horizontal component, often ignored for vertical-only problems such as this.

In problems like this, we often focus on the vertical motion, particularly how energy changes as the object moves upwards. As the stone travels up, its initial kinetic energy is gradually converted into potential energy up to the peak height.
Key aspects of projectile motion include:

• The trajectory depends on the initial speed and angle.
• Gravitational force acts downward, slowing the rise and accelerating the fall.
• A perfect example of the conservation of energy is how potential and kinetic energies exchange roles.

Understanding projectile motion helps appreciate how objects behave under gravity's influence and why conservation principles allow us to predict projectile behavior, such as determining the height where kinetic energy halves.