Problem 481 - JEE Physics

Question

A bomb of mass \(3.0 \mathrm{~kg}\) explodes in air into two pieces of masses \(2.0 \mathrm{~kg}\) and \(1.0 \mathrm{~kg}\). The smaller mass goes at a speed of \(80 \mathrm{~m} / \mathrm{s}\). The total energy imparted to the two fragments is (A) \(1.07 \mathrm{KJ}\) (B) \(2.14 \mathrm{KJ}\) (C) \(2.4 \mathrm{KJ}\) (D) \(4.8 \mathrm{KJ}\)

Step-by-Step Solution

Answer

The total energy imparted to the two fragments is (D) \(4.8 \mathrm{KJ}\).

Step 1: Apply the law of conservation of momentum

The law of conservation of momentum states that the initial momentum of a system must be equal to the final momentum of the system. Initially, the bomb is at rest, so its momentum is 0. After the explosion, it breaks into two pieces with the given masses and velocities. Therefore, we can write the equation: \[0 = m_1v_1 + m_2v_2\] where \(m_1\) and \(m_2\) are the masses of the two fragments, and \(v_1\) and \(v_2\) are their respective velocities.

Step 2: Find the velocity of the larger fragment

We are given the mass and velocity of the smaller fragment (\(m_1 = 1.0 \mathrm{~kg}\), \(v_1 = 80 \mathrm{~m} / \mathrm{s}\)) and the mass of the larger fragment (\(m_2 = 2.0 \mathrm{~kg}\)). We can use these values in the equation from Step 1 to find the velocity of the larger fragment, \(v_2\): \[0 = (1.0 \mathrm{~kg})(80 \mathrm{~m} / \mathrm{s}) + (2.0 \mathrm{~kg})(v_2)\] Solving for \(v_2\), we get: \[v_2 = -\frac{1.0 \cdot 80}{2.0} \mathrm{~m} / \mathrm{s} = -40 \mathrm{~m} / \mathrm{s}\] Notice that the velocity of the larger fragment is negative, indicating that the direction of its motion is opposite to that of the smaller fragment.

Step 3: Calculate the kinetic energy of each fragment

Now we will find the kinetic energy of each fragment, which can be calculated using the formula: \[KE = \frac{1}{2}mv^2\] For the smaller fragment: \[KE_1 = \frac{1}{2}(1.0 \mathrm{~kg})(80 \mathrm{~m} / \mathrm{s})^2 = 3200 \mathrm{~J}\] For the larger fragment: \[KE_2 = \frac{1}{2}(2.0 \mathrm{~kg})(-40 \mathrm{~m} / \mathrm{s})^2 = 1600 \mathrm{~J}\]

Step 4: Calculate the total energy imparted to the fragments

Finally, we add the kinetic energy of each fragment to get the total energy imparted to them during the explosion: \[Total \, Energy = KE_1 + KE_2 = 3200 \mathrm{~J} + 1600 \mathrm{~J} = 4800 \mathrm{~J}\] Since we want the answer in kilojoules (KJ), we can convert it: \[4800 \mathrm{~J} = 4.8 \mathrm{~KJ}\] So, the correct answer is (D) \(4.8 \mathrm{KJ}\).

Key Concepts

Kinetic Energy

Kinetic energy is a fundamental concept in physics that describes the energy an object possesses due to its motion. Whenever an object is moving, it carries kinetic energy, which can be calculated using the formula:

• \( KE = \frac{1}{2}mv^2 \)

Here, \(m\) stands for mass in kilograms, and \(v\) stands for velocity in meters per second. As you can see, kinetic energy is directly proportional to both the mass of the object and the square of its velocity. This means that even slight changes in speed can have a significant impact on the kinetic energy. This aspect notably comes into play when dealing with explosive events, like the one discussed in our exercise.During the explosion of the bomb, both pieces are propelled away from each other, acquiring kinetic energy in the process. We calculated that the smaller fragment (1 kg mass) ends up with a kinetic energy of 3200 Joules while the larger fragment (2 kg mass) ends up with 1600 Joules, giving us a total kinetic energy of 4800 Joules imparted by the explosion.

Physics Problems

Physics problems, particularly those involving conservation laws, often appear challenging but become manageable with step-by-step problem-solving strategies. Let's take some time to break it down into approachable steps. For instance, in our bomb explosion problem, we began by analyzing the system using conservation of momentum. Knowing that the total momentum of the closed system remains constant, we calculated velocities of the individual fragments post-explosion. This foundation is crucial, as it lays the groundwork for determining other quantities, such as kinetic energy. When tackling physics problems, it's essential to:

• Comprehend the core concepts and formulas involved, such as kinetic energy or momentum.
• Start by identifying the known and unknown properties in the problem.
• Apply the proper conservation laws, like momentum, to transition between stages of the problem.
• Step through calculations logically to solve for unknowns, one piece at a time.

Using structured methods and understanding the underlying principles leads to greater success in solving even the most complex physics problems.

Explosion Analysis

Explosion analysis involves examining how energy transformations occur during an explosive event and understanding their consequences on motion. Crucially in explosion analysis, we start by acknowledging the initial condition, which is typically a system at rest. For the exercise given, the initial momentum was 0 since nothing was moving (the bomb was at rest before it exploded). Upon exploding, the bomb splits into two fragments, and the principle of conservation of momentum tells us their combined momentum after the explosion still equals 0. This allows us to calculate the velocity of the unknown fragment, balancing the system's momentum. Following the explosion, each fragment gains kinetic energy, a portion of the total energy released during the explosion. This can be analyzed with:

• Conservation of Momentum: Ensures the total momentum remains constant, even if individual directions change.
• Kinetic Energy Calculations: Quantifies the energy possessed by each fragment in motion.
• Energy Transformation Insights: Understanding how energy initially present in chemical bonds is transformed to motion.

Through explosion analysis, we can predict outcomes and the distribution of energy among fragments, helping in various fields from engineering to safety assessments.