Problem 6 - College Physics

Question

S Show that the kinetic energy of a particle of mass \(m\) is related to the magnitude of the momentum \(p\) of that particle by \(K E=p^{2} / 2 \mathrm{~m}\). (Note: This expression is invalid for particles traveling at speeds near that of light.)

Step-by-Step Solution

Answer

The substitution of the expression for velocity in terms of momentum into the kinetic energy equation yields the desired result, showing that the kinetic energy \(KE\) is indeed related to the momentum \(p\) and mass \(m\) of a particle by the equation \(KE=\frac{p^2}{2m}\)

Step 1: Establish the Expressions for Kinetic Energy and Momentum

Firstly, recall basic definitions in physics. The kinetic energy of a body with mass \(m\) moving with velocity \(v\) is given by \(KE = \frac{1}{2}mv^2\). The momentum of the same body is given by \(p = mv\).

Step 2: Express Velocity in terms of Momentum and Mass

Using the definition of momentum, rearrange the equation to find velocity. By doing this, the equation becomes \(v = \frac{p}{m}\).

Step 3: Substitute for Velocity in Kinetic Energy Expression

Now, substitute the expression for velocity from Step 2 into the kinetic energy equation from Step 1. This gives \(KE = \frac{1}{2}m(\frac{p}{m})^2\). Simplify this to obtain \(KE = \frac{p^2}{2m}\).

Key Concepts

Physics Kinetic Energy

Kinetic energy represents the energy that an object possesses due to its motion. It is a form of energy that's directly related to the motion's velocity and the object's mass. The foundational equation for kinetic energy in classical mechanics is given by \( KE = \frac{1}{2}mv^2 \), where \(m\) represents the object's mass and \(v\) stands for its velocity. This equation implies that the faster an object moves, or the more massive it is, the greater its kinetic energy.

Understanding kinetic energy is crucial because it helps us comprehend how energy is conserved and transferred in a system. For instance, when two objects collide, kinetic energy may be converted into other forms of energy such as heat or sound, but the total energy remains constant. This is a cornerstone of the conservation of energy principle in physics.

When examining problems that involve kinetic energy, it's important to consider both the magnitude of the velocity and the mass of the object. How much kinetic energy an object has is not solely dependent on how fast it's moving, but also on how much substance it has. This distinction is key to understanding the dynamic behaviors of objects in motion.

Momentum

Momentum, often symbolized as \(p\), is a vector quantity that describes the quantity of motion of a moving body. For an object with a fixed mass, momentum is directly proportional to its velocity and is calculated using the formula \(p = mv\), where \(m\) is the mass and \(v\) is the velocity. Momentum is conserved in closed systems, which means that the total momentum before an event, such as a collision, is equal to the total momentum after the event.

This conservation of momentum is a fundamental concept in physics because it helps us predict the outcomes of interactions between moving objects. For example, in a two-car collision, while the speeds and directions may change, the combined momentum of both cars before the collision will equal their combined momentum after collision, assuming no external forces interfere.

One key characteristic of momentum is that it is a vector quantity, which means it has both magnitude and direction. This differs from kinetic energy, which is a scalar quantity, having magnitude only. When solving physics problems, understanding the directional properties of momentum can be essential in determining the system's behavior after interactions that involve motion.

Mass-Velocity Relation

The relationship between an object's mass and its velocity is deeply intertwined with its kinetic energy and momentum. While kinetic energy and momentum involve both mass and velocity, they relate to these quantities in different ways. Momentum is directly proportional to both mass and velocity, implying that if one increases, so does the momentum, given the other quantity remains constant.

Using the relationship \(p = mv\), where \(p\) stands for momentum, \(m\) for mass, and \(v\) for velocity, we see that an object's momentum increases with its velocity. This linear relationship can be visually represented on a graph with a straight line passing through the origin, where the slope of the line corresponds to the object's mass.

In contrast, kinetic energy, described by \(KE = \frac{1}{2}mv^2\), demonstrates that energy increases with the square of velocity. This means that a small increase in velocity results in a larger increase in kinetic energy. The relation between mass and kinetic energy is direct but more complex than with momentum because of the velocity squared term. This relationship is crucial in understanding the dynamic behaviors of objects in various systems, from simple mechanics to more complex systems like orbital mechanics in space.