Question

A radioactive nucleus, initially at rest, decays by emitting an electron and a neutrino at right angles to one another. The momentum of the electron is \(1.2 \times 10^{-22} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\) and that of the neutrino is \(6.4 \times 10^{-23} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\). Find the direction and magnitude of the momentum of the recoiling nucleus.

Short answer

The magnitude of the momentum of the recoiling nucleus is \(1.31 \times 10^{-22} kg\cdot m/s\), and the direction is \(154^{\circ}\) in the negative y direction.

Step-by-step solution

Calculating the Components of the Nucleus' Momentum

Since the electron and neutrino are emitted at right angles, the momentum of the recoiling nucleus should be equal and opposite to the vector sum of their momentums. The x-component of the nucleus' momentum equals the negative of the electron's momentum, and the y-component equals the negative of the neutrino's momentum. Thus, \(p_{x_nucleus} = -1.2 \times 10^{-22} kg\cdot m/s\) and \(p_{y_nucleus} = -6.4 \times 10^{-23} kg\cdot m/s\).

Calculating the Magnitude of the Nucleus' Momentum

The magnitude of a vector can be calculated using Pythagoras theorem. Thus, the magnitude \(p_{nucleus}\) of the momentum of the recoiling nucleus is \(\sqrt{(p_{x_nucleus})^2 + (p_{y_nucleus})^2}\). Substituting the given values gives \(p_{nucleus}\) = \(\sqrt{(-1.2 \times 10^{-22})^2 + (-6.4 \times 10^{-23})^2}\) = \(1.31 \times 10^{-22} kg\cdot m/s\).

Calculating the Direction of the Nucleus' Momentum

The direction of a vector can be calculated using an inverse tangent function to return the counterclockwise angle in radians. Thus, the direction \(\theta_{nucleus}\) of the momentum of the recoiling nucleus is \(tan^{-1}(p_{y_nucleus}/p_{x_nucleus})\). Substituting the given values gives \(\theta_{nucleus}\) = \(tan^{-1}(-6.4 \times 10^{-23} / -1.2 \times 10^{-22})\) = \(2.68 rad\), or 154 degrees in the negative y direction.

Key concepts

Conservation of Momentum

The principle of conservation of momentum is a fundamental concept in physics, stating that if no external forces act on a closed system, the total momentum of that system remains constant.

In the context of radioactive decay, where a nucleus at rest decays into multiple particles, the principle implies that the total vector sum of the momentum of these particles must equal zero, since initially there was no momentum. Picture a calm pond representing the nucleus at rest, and suddenly, three stones are thrown in (the particles after decay) - each stone creates ripples (momentum) in different directions, but the overall stillness (total momentum) of the pond hasn't changed, though it may seem disturbed locally.

Because the ejected electron and neutrino emerged at right angles with known momenta, the recoiling nucleus must therefore exhibit an equal and opposite momentum to comply with this conservation law. Practical applications of this principle extend to understanding the behavior of subatomic particles, as well as in solving collision and explosion problems in classical mechanics.

Momentum

Momentum, often symbolized by the letter 'p', is a measure of the motion of an object and is the product of an object's mass and velocity. It is a vector quantity, possessing both magnitude and direction, which describes the 'oomph' an object carries as it moves.

Imagine a rugby player running – their mass and their speed contribute to their momentum, which would determine how difficult they are to stop. In our problem, the electron and neutrino's momenta are given, and they have momentum because, even as tiny particles, they have mass and speed. These vector quantities are crucial for determining the resultant momentum of the recoiling nucleus. In-depth understanding of how momentum works allows scientists to predict outcomes in particle physics and even space travel, where momentum conservation is paramount.

Vector Sum

A vector sum refers to the combined effect of two or more vectors. It is the total effect that you would witness if all vectors were applied together. Vectors can represent quantities such as momentum, force, or velocity, and are added together using the head-to-tail method, resulting in a resultant vector.

For instance, if two dogs pull on ropes attached to a sled at different angles, the sled will move according to the combined effect of both dogs' pulling forces - this combined effect is the vector sum of the forces. In the given exercise, the momentum vectors of the electron and neutrino must be added using the vector sum method to find the momentum of the recoiling nucleus, ensuring that the direction, not just the magnitude, is taken into account for an accurate result.

Pythagorean Theorem

One of the most well-known theorems in mathematics, the Pythagorean theorem, deals with the lengths of the sides of right-angled triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

The theorem's formula is given by: \( c^2 = a^2 + b^2 \), where 'c' is the hypotenuse, and 'a' and 'b' are the other two sides. In our radioactive decay problem, the theorem helps determine the magnitude of the recoiling nucleus’s momentum vector, by treating the momenta of the electron and neutrino as perpendicular components of a right-angled triangle. The magnitude of the nucleus's momentum is the length of the hypotenuse of this figurative triangle. This geometric approach provides a visual and practical method to solving many physics problems that involve right angles and vector quantities.