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Chapter 26: Problem 52

For a nonrelativistic particle of mass \(m,\) show that \(K=p^{2} /(2 m) .\) [Hint: Start with the nonrelativistic expressions for kinetic energy \(K\) and momentum \(p\).]

Short Answer

Expert verified
Answer: For a nonrelativistic particle of mass m, the relationship between kinetic energy (K) and momentum (p) is given by the equation: \(K=p^{2} /(2 m)\).

Step by step solution

01

Write down the expression for kinetic energy

Kinetic energy (K) for a nonrelativistic particle of mass (m) and velocity (v) is given by the formula: K = \(\frac{1}{2} mv^2\)
02

Write down the expression for momentum

Momentum (p) is defined as the product of an object's mass (m) and its velocity (v): p = mv
03

Solve for velocity in the momentum equation

In order to relate the two expressions, we'll first solve for velocity (v) in the momentum equation above by dividing both sides by mass (m): v = \(\frac{p}{m}\)
04

Substitute the expression for velocity in the kinetic energy equation

Now that we have an expression for velocity (v) in terms of momentum (p) and mass (m), we can substitute it into the kinetic energy equation: K = \(\frac{1}{2}m\left(\frac{p}{m}\right)^2\)
05

Simplify the equation to show K = \(p^2 / (2m)\)

Simplify the above equation to show that the kinetic energy (K) is equal to the square of the momentum (p) divided by twice the mass (2m): K = \(\frac{1}{2}m\left(\frac{p^2}{m^2}\right)\) K = \(\frac{p^2}{2m}\) Thus, we have shown that for a nonrelativistic particle of mass m, \(K=p^{2} /(2 m)\).

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Most popular questions from this chapter

The Tevatron is a particle accelerator at Fermilab that accelerates protons and antiprotons to high energies in an underground ring. Scientists observe the results of collisions between the particles. The protons are accelerated until they have speeds only \(100 \mathrm{m} / \mathrm{s}\) slower than the speed of light. The circumference of the ring is \(6.3 \mathrm{km} .\) What is the circumference according to an observer moving with the protons? [Hint: Let \(v=c-u\) where $v \text { is the proton speed and } u=100 \mathrm{m} / \mathrm{s} .$]
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