Final answer:
An object must move at approximately 1.34 x 10⁸ meters per second in order for its total energy to be 10% greater than its rest energy.
Step-by-step explanation:
The total energy of an object is given by the sum of its rest energy and its kinetic energy. In this case, the question states that the total energy is 10% greater than the rest energy. Let's denote the rest energy as E_rest and the kinetic energy as E_kinetic. The total energy can be expressed as E_total = E_rest + E_kinetic. Since the total energy is 10% greater than the rest energy, we can write E_total = 1.1 × E_rest.
Given that the rest energy of the object is equal to its mass multiplied by the square of the speed of light (E_rest = m × c²), we can substitute this expression into the equation for total energy: 1.1 × m × c² = E_rest + E_kinetic.
Now, we need to find the speed at which the object must move to have a total energy 10% greater than its rest energy. This can be done by equating the kinetic energy to the difference between the total energy and the rest energy: E_kinetic = E_total - E_rest. Substituting the expressions for E_total and E_rest, we get 1.1 × m × c² - m × c² = E_kinetic.
Simplifying the equation, we have 0.1 × m × c² = E_kinetic, which implies that the kinetic energy is equal to 0.1 times the rest mass energy.
Since the kinetic energy can also be expressed as (1/2) × m × v², where m is the mass of the object and v is its velocity, we can equate the two expressions for kinetic energy: 0.1 × m × c² = (1/2) × m × v². Cancelling out the mass from both sides of the equation, we get 0.1 × c² = (1/2) × v².
Solving for v, we have v² = (0.1 × c²) × 2. Taking the square root of both sides of the equation, we get v = √((0.1 × c²) × 2).
Substituting the value of the speed of light, c, which is approximately 3 × 10⁸ meters per second, we can calculate the velocity:
v ≈ √((0.1 × (3 × 10⁸)²) × 2) ≈ √(1.8 × 10¹⁶)
v ≈ 1.34 × 10⁸ meters per second.