A mother has four times the mass of her young son. Both are running with the same kinetic energy. What is the ratio v(son)/v(mother) of their speeds?
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Identify the relationship between kinetic energy, mass, and velocity. The kinetic energy (KE) is given by the formula KE = , where is the mass and is the velocity.
Set up the equation based on the information that the mother has four times the mass of her son. Let the mass of the son be , then the mass of the mother is .
Since both have the same kinetic energy, set their kinetic energies equal to each other: .
Simplify the equation by canceling out the common terms. The equation reduces to .
Take the square root of both sides to solve for the ratio of their velocities: . Thus, the son's speed is twice the speed of the mother.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Kinetic Energy
Kinetic energy (KE) is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. In this scenario, both the mother and son have the same kinetic energy, which implies a relationship between their masses and velocities.
In the context of kinetic energy, if two objects have the same kinetic energy, their masses and velocities are inversely related. Specifically, if one object has a greater mass, it must have a lower velocity to maintain the same kinetic energy as a lighter object. This principle is crucial for determining the speed ratio between the mother and son.
The ratio of speeds (v(son)/v(mother)) can be derived from the relationship between their masses and kinetic energies. Given that the mother has four times the mass of the son, we can express their speeds in terms of their kinetic energies, leading to a straightforward calculation of the speed ratio based on their mass difference.