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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

Two atomic clocks are synchronized. One is put aboard a spaceship that leaves Earth at \(t=0\) at a speed of \(0.750 c .\) (a) When the spaceship has been traveling for \(48.0 \mathrm{h}\) (according to the atomic clock on board), it sends a radio signal back to Earth. When would the signal be received on Earth, according to the atomic clock on Earth? (b) When the Earth clock says that the spaceship has been gone for \(48.0 \mathrm{h},\) it sends a radio signal to the spaceship. At what time (according to the spaceship's clock) does the spaceship receive the signal?
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Particle \(A\) is moving with a constant velocity \(v_{\mathrm{AE}}=+0.90 c\) relative to an Earth observer. Particle \(B\) moves in the opposite direction with a constant velocity \(v_{\mathrm{BE}}=-0.90 c\) relative to the same Earth observer. What is the velocity of particle \(B\) as seen by particle \(A ?\)
Relative to the laboratory, a proton moves to the right with a speed of \(\frac{4}{5} c,\) while relative to the proton, an electron moves to the left with a speed of \(\frac{5}{7} c .\) What is the speed of the electron relative to the lab?
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