Calculating Common Velocity And Energy Loss In A Ball Collision

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This classic physics problem delves into the fundamental principles of collisions, specifically focusing on inelastic collisions where objects stick together after impact. We'll explore how to calculate the common velocity of the combined mass and the energy lost during this process. This is a critical concept in understanding momentum and energy conservation, with applications ranging from car crashes to billiard ball interactions. By breaking down the problem step-by-step, we can gain a deeper understanding of the underlying physics and its real-world implications. Understanding the conservation laws at play allows us to predict the outcome of such collisions and appreciate the transfer of energy and momentum within a system. Furthermore, this analysis is crucial in various fields like engineering, where designing safe and efficient systems often requires understanding how objects interact during collisions. The principles involved, such as the conservation of momentum and the concept of kinetic energy, are foundational in physics and provide a framework for analyzing a wide range of physical phenomena. So, let's embark on this journey to unravel the intricacies of collisions and energy loss.

Understanding the Problem: Collision Dynamics

To tackle this problem effectively, we need to break it down into manageable parts. The core concept here is the conservation of momentum, a fundamental principle in physics. Momentum, often described as "inertia in motion," is the product of an object's mass and its velocity. In a closed system, the total momentum remains constant before and after a collision. This means that the total momentum of the two balls before they collide must equal their combined momentum after they stick together. This is a powerful tool for predicting the motion of objects after a collision. However, it's crucial to note that kinetic energy, while related, is not always conserved in collisions. In inelastic collisions, such as the one described in this problem, some kinetic energy is converted into other forms of energy, like heat or sound, due to the deformation of the objects upon impact. Therefore, we'll also need to calculate the kinetic energy before and after the collision to determine the energy loss. By carefully analyzing the initial conditions (masses and velocities) and applying the conservation laws, we can accurately predict the final velocity and the amount of energy dissipated during the impact. This type of analysis is widely used in various applications, from designing safer vehicles to understanding the behavior of particles in high-energy physics experiments. The problem at hand serves as a great illustration of how physics principles can be applied to solve real-world scenarios.

Step 1: Calculating the Common Velocity

Here, we employ the principle of the conservation of momentum. The formula is as follows:

m1v1 + m2v2 = (m1 + m2)v

Where:

  • m1 = mass of the first ball (5 kg)
  • v1 = velocity of the first ball (10 m/s)
  • m2 = mass of the second ball (6 kg)
  • v2 = velocity of the second ball (-4 m/s) (Note the negative sign, indicating the opposite direction)
  • v = common velocity after the collision (what we want to find)

Now, let's plug in the values:

(5 kg * 10 m/s) + (6 kg * -4 m/s) = (5 kg + 6 kg) * v

50 kg m/s - 24 kg m/s = 11 kg * v

26 kg m/s = 11 kg * v

Now, solve for v:

v = 26 kg m/s / 11 kg

v ≈ 2.36 m/s

Therefore, the common velocity of the balls after the collision is approximately 2.36 m/s in the direction of the first ball's initial motion. This calculation demonstrates the power of the conservation of momentum. By equating the total momentum before and after the collision, we were able to directly determine the final velocity of the combined mass. This principle is universally applicable in closed systems, regardless of the complexity of the collision. The key is to correctly identify the masses and velocities involved and to account for the direction of motion using appropriate signs. This step is crucial for understanding the overall dynamics of the collision and sets the stage for the next step, which involves calculating the kinetic energy loss.

Step 2: Calculating the Loss in Kinetic Energy

Kinetic energy (K.E.) is the energy an object possesses due to its motion and is calculated using the formula:

K.E. = 1/2 * mv^2

First, we calculate the total initial kinetic energy (K.Ei) of the system before the collision:

K.Ei = 1/2 * m1 * v1^2 + 1/2 * m2 * v2^2

K.Ei = (1/2 * 5 kg * (10 m/s)^2) + (1/2 * 6 kg * (-4 m/s)^2)

K.Ei = (1/2 * 5 kg * 100 m^2/s^2) + (1/2 * 6 kg * 16 m^2/s^2)

K.Ei = 250 J + 48 J

K.Ei = 298 J

Next, we calculate the final kinetic energy (K.Ef) of the combined mass after the collision:

K.Ef = 1/2 * (m1 + m2) * v^2

K.Ef = 1/2 * (5 kg + 6 kg) * (2.36 m/s)^2

K.Ef = 1/2 * 11 kg * 5.57 m^2/s^2

K.Ef ≈ 30.64 J

Finally, we calculate the loss in kinetic energy (ΔK.E.) by subtracting the final kinetic energy from the initial kinetic energy:

ΔK.E. = K.Ei - K.Ef

ΔK.E. = 298 J - 30.64 J

ΔK.E. ≈ 267.36 J

Therefore, the loss in kinetic energy during the collision is approximately 267.36 Joules. This substantial loss in kinetic energy highlights the inelastic nature of the collision. The kinetic energy wasn't destroyed; instead, it was transformed into other forms of energy, such as heat generated by the impact and sound waves produced during the collision. Understanding energy transformations is crucial in physics. In this case, the lost kinetic energy manifests as internal energy within the balls and the surrounding environment. This calculation provides a quantitative measure of the energy dissipated during the impact and emphasizes the difference between elastic and inelastic collisions.

Conclusion: The Physics of Collisions

In summary, we've successfully calculated the common velocity of the two balls after the collision and determined the loss in kinetic energy. The common velocity was found to be approximately 2.36 m/s, and the loss in kinetic energy was about 267.36 Joules. This problem exemplifies the application of fundamental physics principles, namely the conservation of momentum and the concept of kinetic energy. The collision was inelastic, as evidenced by the significant loss in kinetic energy, which was converted into other forms of energy. The calculations performed are essential for understanding the dynamics of collisions and are applicable in various fields, from engineering design to sports analysis. The principles illustrated here extend beyond simple collisions and form the basis for understanding more complex interactions in the physical world. By analyzing these types of problems, we can gain a deeper appreciation for the elegance and power of physics in explaining the phenomena around us. Furthermore, this example underscores the importance of considering both momentum and energy when analyzing collisions. While momentum is always conserved in a closed system, kinetic energy may or may not be, depending on the nature of the collision.