Question

An electron, mass \(m\), collides head-on with an atom, mass \(M\), initially at rest. As a result of the collision, a characteristic amount of energy \(E\) is stored internally in the atom. What is the minimum initial speed \(v_{0}\) that the electron must have? (Hint: Conservation principles lead to a quadratic equation for the final electron speed \(v\) and a quadratic equation for the final atom speed \(V\). The minimum value, \(v_{0}\), follows from the requirement that the radical in the solutions for \(v\) and \(V\) be real.)

Short answer

The minimum initial speed \(v_{0}\) of the electron can be obtained by solving the quadratic equation set by conservation laws and ensuring the discriminant is non-negative. The mathematical steps will depend on the values for \(m\), \(M\), and \(E\), which are not given.

Step-by-step solution

Application of Conservation of Linear Momentum

The initial momentum is entirely due to the electron, \(mv_{0}\), and the final momentum is the sum of the momenta of the electron and the atom, \(mv + MV\). Applying conservation of linear momentum, the equation can be written as: \(mv_{0}=mv+MV\). This equation can be rearranged to get: \(MV = mv_{0}-mv\).

Application of Conservation of Energy

Before the collision, the kinetic energy of the system is solely due to the electron, and it is given by \(\frac{1}{2}m{v_{0}}^{2}\). After the collision, kinetic energy is associated with both the electron and the atom and there is energy stored in the atom. Energy conservation law leads to: \(\frac{1}{2}m{v_{0}}^{2}=\frac{1}{2}m{v}^{2} + \frac{1}{2}M{V}^{2} + E\). Substitute value of MV from step 1 in this equation to get: \(\frac{1}{2}m{v_{0}}^{2}=\frac{1}{2}m{v}^{2} + \frac{1}{2}M[mv_{0}-mv]^{2}/M + E\). Simplify it to find the quadratic equation for v.

Find Minimum Initial Speed

To have real solutions for \(v\) and \(V\), the discriminant of the quadratic equation must be equal or greater than zero. After solving the quadratic and condition for real roots(i.e., discriminant >= 0), we get minimum value of \(v_{0}\) which provides the solution to our question.

Key concepts

Linear Momentum Conservation

One of the most fundamental principles in physics is the conservation of linear momentum. It dictates that in an isolated system (where no external forces are acting), the total momentum remains constant throughout any process. This means that the momentum before an event is equal to the momentum after it.

Let's consider a simple collision, like the one described in the exercise. An electron, with a certain velocity, collides with an atom at rest. The electron's initial momentum is the product of its mass and its initial speed, represented by the expression \(m v_0\). After they collide, the electron and the atom both move, and their combined momentum must equal that of the electron before the collision. This is written as \(m v_0 = m v + M V\), where \(m\) and \(M\) are the masses of the electron and atom respectively, and \(v\) and \(V\) are their speeds after the collision.

The beauty of this concept is its universal application; momentum conservation can be seen in everything from playground seesaws to the vast motions of celestial bodies in space.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. The faster something moves, the more kinetic energy it has, and this relationship can be quantified by the equation \(\frac{1}{2}mv^2\), where \(m\) is the mass of the object, and \(v\) is its velocity.

In our exercise, initially, the kinetic energy is all in the moving electron. After the collision, however, both the electron and the atom have some kinetic energy since they're both moving. Additionally, some energy becomes internal energy within the atom, noted as \(E\). We account for all forms of energy before and after the collision using the conservation of energy principle, which helps us understand how this kinetic energy is distributed among the objects after the collision.

Elastic Collision

An elastic collision is an event where the total kinetic energy and total linear momentum of the system are conserved. During such collisions, no kinetic energy is lost to sound, heat, or deformation. In the idealized situation of our exercise, however, we're taking into account an additional energy \(E\) that is stored in the atom, which indicates that the collision is not perfectly elastic because some kinetic energy is converted into internal energy.

In context, considering a partially elastic collision allows us to apply conservation laws realistically since in the real world, some energy is often transformed rather than strictly conserved as kinetic energy. This subtle difference affects the outcome and is essential for solving the problem accurately.

Quadratic Equation

A quadratic equation is a second-order polynomial equation, typically in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. These equations are fundamental in algebra and appear frequently in various physical scenarios, including motion under constant acceleration and, as in our exercise, solving for variables in conservation equations.

In physics problems, finding the real roots of a quadratic is often essential. The requirement that the roots must be real (not imaginary) imposes certain conditions on the values of \(a\), \(b\), and \(c\): specifically, the discriminant \(b^2 - 4ac\) must be greater than or equal to zero. When we apply this to the conservation laws in our collision problem, we find the minimum speed needed for the electron before the collision to ensure that the resulting speeds of both the electron and atom are real and physically meaningful. This highlights the deep interconnection between algebra and physics.