Final answer:
To find the original speeds of the father and the son, we use the kinetic energy formula and the given relationships between their masses and energies. Setting up equations based on these conditions and solving for the velocities yields the answers.
Step-by-step explanation:
To solve for the original speeds of the father and the son, we first need to understand that kinetic energy is given by the equation KE = (1/2)mv2, where 'm' is the mass and 'v' is the velocity of the object. With the father having 1/3 the kinetic energy of the son and 1/3 the mass, and after increasing his speed by 13 m/5 to match the son's kinetic energy, we can set up equations to solve for their original speeds.
Let vf and vs be the original velocities of the father and the son respectively. The father's mass is 3 times that of the son, let's say son's mass is 'm'. The father's kinetic energy is (1/2)(3m)vf2 and the son's kinetic energy is (1/2)mvs2. According to the problem, the father's kinetic energy is 1/3 of the son's, so we have (1/2)(3m)vf2 = (1/3)(1/2)mvs2. Simplifying the equation, we find vf = vs/31/2.
After the father speeds up by 13 m/5, their kinetic energies are the same, so: (1/2)(3m)(vf + 13/5)2 = (1/2)mvs2. Using the first equation to eliminate vf in terms of vs and solving this equation gives us the original velocities of the father and the son.