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Chapter 14: Problem 16

The big difference between the equations for momentum and kinetic energy is that one is proportional to \(v\) and one to \(v^{2} .\) Both, however, are proportional to \(m\). Suppose someone tells you that there's a third quantity, funkosity, defined as \(f=m^{2} v\), and that funkosity is conserved. How do you know your leg is being pulled?

Short Answer

Expert verified
Funkosity can't be conserved because it depends on \( m^2 \), which doesn't change in isolated systems, unlike velocities. It's an unrealistic, non-physical concept.

Step by step solution

01

Understand Momentum and Kinetic Energy

Momentum is defined as the product of mass and velocity, i.e., \( p = mv \), which means momentum is directly proportional to velocity \( v \) and mass \( m \). Kinetic energy, on the other hand, is defined as \( KE = \frac{1}{2} mv^2 \), making it proportional to the mass \( m \) and the square of the velocity \( v^2 \).
02

Analyze the Definition of Funkosity

Funkosity is proposed as \( f = m^2 v \), which makes it proportional to the square of the mass \( m^2 \) and velocity \( v \). This contrasts momentum and kinetic energy as it involves the square of the mass.
03

Explore Conservation Conditions

For a quantity to be conserved, it must remain constant in a closed system over time, regardless of interactions. Momentum and kinetic energy have their conservation principles derived from Newton's laws and energy conservation laws. To be conserved, funkosity would also need adherence to such physical laws.
04

Identify Why Funkosity Cannot Be Conserved

As funkosity depends on \( m^2 \), consider a system of two colliding objects. For funkosity to remain conserved after collision, the mass squared term introduces discrepancy as masses typically don't change in isolation or collisions, unlike velocity which can easily redistribute to conserve momentum and energy. This mass squared term implies non-physical quantities since mass additions/subtractions aren't realistic in closed, isolated systems.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Momentum
Momentum is a fundamental concept in physics. It is defined as the product of an object's mass and its velocity, expressed by the formula \( p = mv \). This equation shows that momentum is directly proportional to both mass \( m \) and velocity \( v \). When either the mass or velocity increases, the momentum also increases.

It plays a crucial role in the conservation of momentum principle, which states that the total momentum of a closed system is constant, provided no external forces act on it.
  • Momentum is a vector quantity, meaning it has both magnitude and direction.
  • In collisions, momentum can be transferred between objects, ensuring total momentum is conserved.
Kinetic Energy
Kinetic energy is another vital concept in physics, representing the energy an object has due to its motion. The formula for kinetic energy is \( KE = \frac{1}{2} mv^2 \). Here, it is apparent that kinetic energy is proportional to the mass \( m \) and the square of the velocity \( v^2 \).

Because of this velocity squared factor, a small increase in speed results in a significant increase in kinetic energy. When considering conservation, kinetic energy is preserved in elastic collisions.
  • Kinetic energy can be transformed, such as into potential energy, demonstrating the conservation of energy principle.
  • Unlike momentum, kinetic energy is a scalar quantity, meaning it doesn鈥檛 have direction, only magnitude.
Mass Velocity Relationship
The relationship between mass and velocity is crucial in understanding momentum and kinetic energy. Momentum features a direct relationship with velocity, where momentum increases linearly as velocity increases.

Kinetic energy, on the other hand, has a quadratic relationship with velocity. As velocity increases, kinetic energy increases exponentially, due to the \( v^2 \) term in the kinetic energy formula.
  • This quadratic dependency means a doubling of velocity results in quadrupling the kinetic energy.
  • This relationship also highlights why faster-moving objects possess much higher kinetic energy compared to slower ones with the same mass.
Physical Quantities Conservation
The conservation laws for physical quantities ensure that certain properties remain constant over time within a closed system. Momentum conservation is a direct outcome of Newton's laws.

For momentum to be conserved, the system must be closed, with no external forces acting upon it. Kinetic energy conservation, however, holds in elastic collisions.
  • In any closed system, total momentum before and after interactions remains the same.
  • Conservation of energy implies that the total energy within an isolated system remains constant, only transforming between types.
  • Unlike proposed funkosity, valid conservation laws do not require physically unrealistic conditions such as squaring mass.

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Most popular questions from this chapter

A rocket ejects exhaust with an exhaust velocity \(u\). The rate at which the exhaust mass is used (mass per unit time) is \(b\). We assume that the rocket accelerates in a straight line starting from rest, and that no external forces act on it. Let the rocket's initial mass (fuel plus the body and payload) be \(m_{i}\), and \(m_{f}\) be its final mass, after all the fuel is used up. (a) Find the rocket's final velocity, \(v\), in terms of \(u, m_{i}\), and \(m_{f} .\) Neglect the effects of special relativity. (b) A typical exhaust velocity for chemical rocket engines is 4000 \(\mathrm{m} / \mathrm{s} .\) Estimate the initial mass of a rocket that could accelerate a one-ton payload to \(10 \%\) of the speed of light, and show that this design won't work. (For the sake of the estimate, ignore the mass of the fuel tanks. The speed is fairly small compared to \(c\), so it's not an unreasonable approximation to ignore relativity.)

A bullet leaves the barrel of a gun with a kinetic energy of 90 J. The gun barrel is 50 cm long. The gun has a mass of 4 kg, the bullet 10 g. (a) Find the bullet鈥檚 final velocity. (b) Find the bullet鈥檚 final momentum. (c) Find the momentum of the recoiling gun. (d) Find the kinetic energy of the recoiling gun, and explain why the recoiling gun does not kill the shooter.

When the contents of a refrigerator cool down, the changed molecular speeds imply changes in both momentum and energy. Why, then, does a fridge transfer power through its radiator coils, but not force?

Two blobs of putty collide head-on and stick. The collision is completely symmetric: the blobs are of equal mass, and they collide at equal speeds. What becomes of the energy the blobs had before the collision? The momentum?

Show that for a body made up of many equal masses, the equation for the center of mass becomes a simple average of all the positions of the masses.
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