Ch 09: Work and Kinetic Energy

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 09: Work and Kinetic Energy

Problem 3

Chapter 9, Problem 3

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?

Verified step by step guidance

Start by recalling the formula for kinetic energy:

K = \frac{1}{2} m v^{2}

, where

K

is the kinetic energy,

m

is the mass, and

v

is the velocity.

Since the mother and son have the same kinetic energy, set their kinetic energy expressions equal to each other:

\frac{1}{2} m v_{son} v^{2} = \frac{1}{2} (4 m) v_{mother} v^{2}

, where the mother's mass is four times the son's mass.

Cancel out the common terms, such as

\frac{1}{2}

and

m

, from both sides of the equation. This simplifies to:

{v_{son}}^{2} = 4 {v_{mother}}^{2}

Take the square root of both sides to solve for the ratio of their speeds:

\frac{v_{son}}{v_{mother}} = \sqrt{4}

Simplify the square root to find the ratio:

\frac{v_{son}}{v_{mother}} = 2

. Thus, the son's speed is twice the mother's speed.

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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 that can be analyzed to find their speed ratio.

Mass and Velocity Relationship

In physics, when 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 in the given problem.

Ratio of Speeds

The ratio of speeds (v(son)/v(mother)) can be derived from the relationship between their masses and kinetic energies. Since the mother has four times the mass of her son and both have equal kinetic energy, we can set up a proportion based on their kinetic energy equations to find the exact ratio of their speeds, illustrating how mass affects velocity in motion.

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A mother has four times the mass of her young son. Both are running with the same kinetic energy. What is the ratio vson/vmother of their speeds?

Verified video answer for a similar problem:

Key Concepts

Kinetic Energy

Mass and Velocity Relationship

Ratio of Speeds

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?