To find the equation of the path for a particle with central acceleration that varies inversely to the cube of the distance, we can use the formula for centripetal acceleration and compare it with the given information. We can solve the equation of the path by equating the centripetal acceleration to the equation for central acceleration and rearranging for r in terms of known variables. The equation to the path is r = 4v₀⁴/a, where v₀ is the velocity for a circle of radius a.
To find the equation of the path for a particle with central acceleration that varies inversely as the cube of the distance, we need to first understand the concepts involved. The central acceleration is the acceleration that acts toward the center of a circular path, and it is given by the formula ac = v²/r, where v is the velocity and r is the radius. In this case, the central acceleration varies inversely to the cube of the distance, which means that ac ∝ 1/r³.
Additionally, we are given that the particle is projected from an apse, which means it is at its greatest distance from the origin, which is a. The velocity of the particle is √2 times the velocity for a circle of radius a.
To find the equation of the path, we can start by using the formula for centripetal acceleration and comparing it with the given information. Since the central acceleration is inversely proportional to the cube of the distance, we can write ac = k/r³, where k is a constant.
Using the formula v = √(ar), where v is the velocity and r is the radius, we can solve for v in terms of r. Given that the velocity for a circle of radius a is v₀, we can write √2v₀ = √(ak/a). Solving for a, we get a = (√2v₀)²/ak = 2v₀²/ak, where k is the same constant from the previous equation.
Substituting this value of a into the equation for ac, we have ac = k/(2v₀²/ak)³ = k/(8v₀⁶/a³k³) = ak⁴/(8v₀⁶). Equating this to the formula for centripetal acceleration, ac = v²/r, we can solve for r in terms of k and v₀. Rearranging the equation, we get r = (8v₀⁶)/(ak⁴) = (8v₀⁶)/(2v₀²) = 4v₀⁴/a.
Therefore, the equation to the path is r = 4v₀⁴/a, where v₀ is the velocity for a circle of radius a.