Problem 60 - College Physics

Question

An unstable nucleus of mass 1.7 3 10226 kg, initially at rest at the origin of a coordinate system, disintegrates into three particles. One particle, having a mass of m1 5 5.0 3 10227 kg, moves in the positive y- direction with speed v1 5 6.0 3 106 m/s. Another particle, of mass m2 5 8.4 3 10227 kg, moves in the positive x- direction with speed v2 5 4.0 3 106 m/s. Find the magnitude and direction of the velocity of the third particle.

Step-by-Step Solution

Answer

The magnitude of the velocity of the third particle is \(v3 = p3/m3\), and the direction can be found by resolving the momentum components into x and y directions. Use trigonometric functions for finding the direction.

Step 1: Calculate the mass of the third particle

Subtract the sum of the masses of the first and second particles from the mass of the nucleus. So, \(m3 = m - (m1 + m2) = 1.7*10^{26} - (5.0*10^{27} + 8.4*10^{27}) = 1.1*10^{26} kg\)

Step 2: Calculate the momentum of the first and second particles

Momentum (p) is given by \(p = mv\). So, for first particle \(p1 = m1*v1 = 5.0*10^{27}*6.0*10^{6} = 3.0*10^{34} Kg*m/s \) along +y-axis ; for the second particle \( p2 = m2*v2 = 8.4*10^{27}*4.0*10^{6} = 3.36*10^{34} Kg*m/s \) along +x-axis

Step 3: Calculate the momentum of the third particle

The momentum of the third particle will be equal in magnitude but opposite in direction (let's say -x and -y are the directions) to the vector sum of momentums of the first and second particles, i.e., \( p3 = -(p1 + p2) \)

Step 4: Find the magnitude and direction of the velocity of the third particle

The magnitude of velocity of third particle is given by \(v3 = p3/m3\). The direction of the third particle can then be found using trigonometric functions or from a vector triangle.

Key Concepts

Nuclear Disintegration

Nuclear disintegration refers to the breakup of an atomic nucleus into smaller particles. In the given problem, an unstable nucleus initially at rest splits into three particles. This process is a spontaneous event where the original nucleus loses its stability and divides. Understanding nuclear disintegration is essential, as it influences the conservation laws, such as the conservation of momentum.

A few key points about this concept include:

• The sum of the masses after disintegration is less than the original nucleus mass, due to a reduction in system mass caused by energy release.
• Each resulting particle from the disintegration has its mass and velocity.
• The total momentum before disintegration should equal the total momentum after it, to obey the law of conservation of momentum.

In this scenario, the third particle has its characteristics determined by the behavior of the first two, ensuring the total momentum remains the same upon disintegration. This highlights how nuclear disintegration results in a balanced yet dynamic system.

Momentum Calculation

Momentum is a measure of the motion in an object, defined as the product of mass and velocity. When a body splits into different parts, as in nuclear disintegration, understanding how to calculate momentum is crucial.

Here's how to approach it:

• The momentum of each particle is calculated by multiplying its mass by its speed (velocity).
• For conservation purposes, the total momentum of the system after disintegration must equal the initial momentum, which is zero in this case since the nucleus was at rest.
• For the first particle, the momentum along the y-direction can be calculated as \( p_1 = m_1 \times v_1 \).
• For the second particle, solve for momentum along the x-direction: \( p_2 = m_2 \times v_2 \).

In the exercise, the given masses and velocities allow us to compute these momentums directly. In physics, mastering the determination and conservation of momentum helps solve plenty of dynamic problems effectively.

Vector Addition

Vector addition is a mathematical operation that combines vectors, often used to determine resulting directions and magnitudes in physics problems like this one. Understanding vector addition is fundamental to finding how a third particle behaves in terms of its momentum and velocity.

When dealing with vectors:

• Vectors have both magnitude and direction.
• The momentum vectors for each particle are perpendicular to each other in the problem, with one along the x-axis and the other along the y-axis.
• To compute the resultant vector for the third particle, use vector addition of momentums: \( p_3 = -(p_1 + p_2) \).
• The direction is determined using trigonometric relationships, generally through tangent and arctan functions.

By applying vector addition, you combine the two momentums and get the third one, allowing determination of how the momentum is distributed post-disintegration. This process elucidates the coupling of direction and magnitude inherent in analyzing motion outcomes.