Final answer:
The ratio of their speeds is about 1:√3.
Based on the kinetic energy formula, the mother, with three times the mass of her son and both having the same kinetic energy, has a speed that is approximately 0.577 times the speed of her son.
Step-by-step explanation:
The question involves kinetic energy and mass-speed relationship in the context of physics. Kinetic energy (KE) is given by the formula KE = (1/2)mv², where m is the mass of an object and v is its velocity.
If two objects have the same kinetic energy, we can use the kinetic energy formula to express the relationship between their masses and speeds.
Let's denote the mother's mass as 3m and the son's mass as m, given that the mother has three times the mass of her son.
Because both have the same kinetic energy, their equations for kinetic energy would be:
(1/2)(3m)v₁² = (1/2)mv₂²
Where v₁ is the mother's speed and v₂ is the son's speed.
By simplifying this equation, we find that v₁²/v₂² = 1/3.
Taking the square root of both sides gives us:
v₁/v₂ = √(1/3)
This means that the mother's speed is √1/√3 times the speed of her son.
Since √3 is approximately 1.732, the mother is running at roughly 0.577 times the speed of her son.
The final ratio of their speeds, v₁:v₂, is therefore approximately 1:√3.