Question

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

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

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 = \(\frac{1}{2} mv^2\)

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

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 = \(\frac{p}{m}\)

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 = \(\frac{1}{2}m\left(\frac{p}{m}\right)^2\)

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, \(K=p^{2} /(2 m)\).