Question

If a skater of weight \(4 \mathrm{~kg}\) has initial speed \(4 \mathrm{~m} / \mathrm{s}\) and \(2^{\text {nd }}\) one of weight \(6 \mathrm{~kg}\) has \(6 \mathrm{~m} / \mathrm{s}\). After collision, they have speed (couple) \(6 \mathrm{~m} / \mathrm{s}\). Then the loss in \(\mathrm{K} . \mathrm{E}\). is..... (A) \(48 \mathrm{~J}\) (B) zero (C) \(96 \mathrm{~J}\) (D) None of these

Short answer

The loss in Kinetic Energy is -40 J, which indicates an increase in Kinetic Energy by 40 J. Hence, the correct option is (D) None of these.

Step-by-step solution

Calculate the initial kinetic energy of each skater

We can find the kinetic energy of each skater using the formula for kinetic energy: K.E. = \(0.5 \cdot m \cdot v^2\), where m is the mass of the skater, and v is the velocity of the skater. For skater 1: Mass m1 = 4 kg Velocity v1 = 4 m/s K.E.1 = 0.5 * m1 * \(v1^2\) = 0.5 * 4 * \(4^2\) = 32 J For skater 2: Mass m2 = 6 kg Velocity v2 = 6 m/s K.E.2 = 0.5 * m2 * \(v2^2\) = 0.5 * 6 * \(6^2\) = 108 J

Calculate the final kinetic energy of the couple

Now let's calculate the final K.E. of the couple after the collision. We need to find their combined mass and use the couple's velocity mentioned in the problem. Total mass of the couple, M = m1 + m2 = 4 kg + 6 kg = 10 kg Velocity of the couple, V = 6 m/s Final K.E. =0.5 * M * \(V^2\) = 0.5 * 10 * \(6^2\) = 180 J

Calculate the loss in kinetic energy

To find the loss in kinetic energy, we subtract the final K.E. from the sum of the initial K.E.s of each skater: Loss in K.E. = (K.E.1 + K.E.2) - Final K.E. Loss in K.E. = (32 J + 108 J) - 180 J = 140 J - 180 J = -40 J Since the loss in K.E. is negative, this means that there is actually an increase in the K.E. of 40 J. Therefore, the correct answer is: (D) None of these.

Key concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. Think of it as the energy of movement. It can be calculated using the formula: \[\text{K.E.} = \frac{1}{2} \cdot m \cdot v^2\]where:

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

The formula shows that kinetic energy depends on both how heavy (\(m\) mass) the object is and how fast (\(v\) velocity) it's moving. The faster something moves and the heavier it is, the more kinetic energy it will have.
For instance, in our exercise:

• Skater 1, who is lighter, has a kinetic energy of 32 J.
• Skater 2, who moves faster with more mass, has a kinetic energy of 108 J.

Kinetic energy plays a crucial role in understanding how energy is transferred and transformed during collisions.

Mass and Velocity

Mass and velocity are two fundamental concepts in physics that greatly affect kinetic energy. Mass refers to how much matter is in an object, and it is measured in kilograms (kg). Velocity is a measure of how fast an object is moving in a specific direction, measured in meters per second (m/s). These two properties work together in the kinetic energy formula to determine how much energy an object in motion has. For example, in our scenario:

• Skater 1 has a mass of 4 kg and moves at a velocity of 4 m/s.
• Skater 2 has a mass of 6 kg and moves at 6 m/s.

Both skaters contribute to the total energy of the system, but because kinetic energy increases with the square of the velocity, the speed of an object has a larger impact on kinetic energy than its mass. This is why even small increases in speed can result in large changes in kinetic energy.

Energy Conservation

Energy conservation is a key principle in physics, stating that energy cannot be created or destroyed, but only transformed from one form to another. Understanding this concept helps explain what happens in a system after a collision. In the exercise, after both skaters collide and couple together, we calculate their final kinetic energy. This final energy is 180 J, which is greater than the sum of the initial energies of 140 J. This increase appears counter-intuitive but indicates the system has gained energy due to an external source of work or energy transfer not considered in basic calculations. In a closed system with no external forces, the total energy before and after a collision should remain constant. Any changes in kinetic energy imply energy was transferred to or from other forms, such as potential energy or through external forces. This demonstrates how energy can be redistributed within a system, aligning with the conservation law, and highlights the importance of considering all energy interactions in a full analysis.