Question

If a proton and an alpha particle (composed of two protons and two neutrons) are each accelerated from rest through the same potential difference, how do their resulting speeds compare? a) The proton has twice the speed of the alpha particle. b) The proton has the same speed as the alpha particle. c) The proton has half the speed of the alpha particle. d) The speed of the proton is \(\sqrt{2}\) times the speed of the alpha particle. e) The speed of the alpha particle is \(\sqrt{2}\) times the speed of the proton.

Short answer

Answer: e) The speed of the alpha particle is √2 times the speed of the proton.

Step-by-step solution

Write down the conservation of energy principle

When a charged particle is accelerated through a potential difference, its potential energy is converted into kinetic energy. This is represented by the following formula: \(eV = \frac{1}{2}mv^2\) Where \(e\) is the charge of the particle, \(V\) is the potential difference, \(m\) is the mass of the particle, and \(v\) is its final speed. Note that \(eV\) represents the energy gained by the particle due to the potential difference.

Set up equations for the proton and the alpha particle

Write down the energy conservation equations for the proton and the alpha particle separately: For the proton: \(eV = \frac{1}{2}m_pv_p^2\) For the alpha particle: \(2eV = \frac{1}{2}m_\alpha{v_\alpha}^2\) Here, \(m_p\) is the mass of the proton and \(v_p\) is its final speed, \(m_\alpha\) is the mass of the alpha particle, \(v_\alpha\) is its final speed, and the charge of the alpha particle is \(2e\) since it contains two protons.

Solve for the ratio of their speeds

Divide the energy conservation equation for the proton by the energy conservation equation for the alpha particle: \(\frac{eV}{2eV} = \frac{\frac{1}{2}m_pv_p^2}{\frac{1}{2}m_\alpha{v_\alpha}^2}\) This simplifies to: \(\frac{1}{2} = \frac{m_p}{m_\alpha}\frac{v_p^2}{{v_\alpha}^2}\) We know that the alpha particle has twice the mass of the proton and twice the charge. The mass of an alpha particle is \(4m_p\), so we can rewrite the equation as: \(\frac{1}{2} = \frac{m_p}{4m_p}\frac{v_p^2}{{v_\alpha}^2}\)

Solve for the speed of each particle

Now, solve the equation for the ratio of the speeds: \({v_\alpha}^2 = 2v_p^2\) Taking the square root of both sides: \(v_\alpha = \sqrt{2}v_p\) So, the correct answer is: e) The speed of the alpha particle is \(\sqrt{2}\) times the speed of the proton.

Key concepts

Conservation of Energy in Physics

The conservation of energy is a fundamental principle in physics, stating that energy cannot be created or destroyed; it can only change from one form to another. This concept is pivotal in understanding particle acceleration, where potential energy (energy due to position in a field) is converted into kinetic energy (energy of motion). When a charged particle, like a proton or an alpha particle, is accelerated by an electric field through a potential difference, its potential electrical energy decreases as it moves, and this energy is not lost but transformed into kinetic energy. As per the conservation law, the total energy before and after the acceleration remains constant, leading to the equation:

\[ eV = \frac{1}{2}mv^2 \]
where \(e\) stands for the electrical charge of the particle, \(V\) is the potential difference, \(m\) is the particle's mass, and \(v\) is the final velocity. This conversion highlights the direct interplay between the electric potential energy due to the potential difference and the kinetic energy gained by the particle.

Kinetic Energy of Charged Particles

The kinetic energy (\(E_k\) of charged particles such as protons and alpha particles is determined by their mass and the speed at which they are traveling. The formula for kinetic energy is:

\[ E_k = \frac{1}{2}mv^2 \]
For charged particles being accelerated through the same potential difference, as mentioned in the exercise, their kinetic energy will depend on both their charge and mass. Since the alpha particle has a charge that is twice that of a proton and a correspondingly higher mass (due to it being composed of two protons and two neutrons), when both particles are accelerated through the same potential difference, the amount of energy they gain is also different. Specifically, the alpha particle gains twice as much energy because of its double charge, evidencing that the kinetic energy has a direct correlation with the particle's charge and potential difference it traverses.

Speed Comparison of Protons and Alpha Particles

When comparing the speeds of a proton and an alpha particle accelerated through the same potential difference, we need to consider their mass and charge. An alpha particle contains more mass than a proton — actually, four times more, given it contains two protons and two neutrons. Coupled with this is the fact that the alpha particle also has twice the charge of a proton.

To predict how their speeds compare, we must understand that, because the alpha particle is heavier, it requires more energy to achieve the same speed as a proton. However, it also gains more energy from the same potential difference due to its higher charge. Mathematically, when the equations for kinetic energy are set up and compared for both particles, we find a distinct relationship between their speeds. The proton, being lighter, tends to move faster, yet the dual charge of the alpha particle with its accompanying mass results in a trade-off that leaves the alpha particle moving at a speed that is \(\sqrt{2}\times\) that of the proton, a balance rooted in the conservation of energy and differences in mass and charge.