Final answer:
The rest energy of the alpha particle is 5.76×10⁻¹⁰ joules. It is not okay to use the classical formula for kinetic energy at high speeds. The kinetic energy of the alpha particle is 2.01 joules.
Step-by-step explanation:
(a) What is its rest energy?
The rest energy of an object can be calculated using the equation E = mc², where E is the energy, m is the mass, and c is the speed of light.
Using the given mass of the alpha particle (6.40×10⁻²⁷ kg) and the speed of light, calculate the rest energy.
Rest Energy (E) = (Mass (m) * Speed of Light (c) * Speed of Light (c))
Rest Energy (E) = (6.40×10⁻²⁷ kg * 3.00×10⁸ m/s * 3.00×10⁸ m/s)
Rest Energy (E) = 5.76×10⁻¹⁰ joules
(b) Is it okay to calculate its kinetic energy using the formula 12mv²?
No, it is not okay to use the formula 12mv² to calculate the kinetic energy at high speeds. The formula 12mv² is the classical formula for kinetic energy. At high speeds, relativistic effects need to be taken into account and the correct formula to use is the relativistic kinetic energy formula.
(c) What is its kinetic energy?
The relativistic kinetic energy formula is given by K = (γ - 1)mc², where K is the kinetic energy, γ is the Lorentz factor (1 / √(1 - v²/c²)), m is the mass, and c is the speed of light.
Using the given speed of the alpha particle (0.9992 times the speed of light) and the mass, calculate the kinetic energy.
Lorentz factor (γ) = 1 / √(1 - (v/c)²)
Lorentz factor (γ) = 1 / √(1 - (0.9992)²)
Lorentz factor (γ) = 5.01
Kinetic Energy (K) = (Lorentz factor (γ) - 1) * Mass (m) * Speed of Light (c) * Speed of Light (c)
Kinetic Energy (K) = (5.01 - 1) * 6.40×10⁻²⁷ kg * 3.00×10⁸ m/s * 3.00×10⁸ m/s
Kinetic Energy (K) = 2.01 joules
(d) Which is true?
Based on the information given, the rest energy of the alpha particle is 5.76×10⁻¹⁰ joules and its kinetic energy is 2.01 joules. It is true that the kinetic energy is significantly greater than the rest energy for a particle moving at high speeds.