Kinetic Energy — Question 4

Question 4

A speed uncertainty produces asymmetric energy bounds

Use classical translational kinetic energy $K=\tfrac12 m|\vec v|^2$. A particle has mass in $[1.9,2.1]$ kg and speed in $[4.9,5.1]$ m/s. Every combination is allowed. A report gives the central estimate 25 J.

1. Find the exact allowed kinetic-energy interval.

2. Compare the upper and lower deviations from 25 J.

3. Could a separately reported energy of 28 J agree with the stated hard bounds?

Question 4 - Solution

A speed uncertainty produces asymmetric energy bounds

(a) Use monotonicity for positive mass and speed.

The governing relation is $$\boxed{\frac{\partial K}{\partial m}=\tfrac12v^2>0,\qquad\frac{\partial K}{\partial v}=mv>0}$$

$$K_{\min}=\tfrac12(1.9)(4.9)^2=22.8095\ \mathrm J,\qquad K_{\max}=\tfrac12(2.1)(5.1)^2=27.3105\ \mathrm J.$$

(b) Subtract the central estimate.

$$25-K_{\min}=2.1905\ \mathrm J,\qquad K_{\max}-25=2.3105\ \mathrm J.$$ The quadratic speed dependence produces unequal downward and upward deviations; independent bounds do not define a statistical error.

(c) Test intersection, not plausibility.

No. An exact 28 J lies above 27.3105 J. At least one bound, measurement or modeling assumption would need revision. The intervals are not probabilistic error bars.