Linear Momentum — Question 7

Question 7

Hard bounds on a reconstructed velocity

A positive mass lies within $1.9\le m\le2.1\,\mathrm{kg}$. Its signed momentum is bounded by $9.8\le p_x\le10.2\,\mathrm{kg\,m/s}$. Treat these as independent hard bounds.

1. Find the sharp velocity interval.

2. Explain why dividing both lower bounds and both upper bounds fails to give the full interval.

Question 7 - Solution

Hard bounds on a reconstructed velocity

(a) Derivation and calculation

For positive mass and momentum, $$v_x=\frac{p_x}{m},\qquad\frac{\partial v_x}{\partial p_x}=\frac1m>0,\quad \frac{\partial v_x}{\partial m}=-\frac{p_x}{m^2}<0.$$ $$\boxed{\frac{9.8}{2.1}\le v_x\le\frac{10.2}{1.9}} \Rightarrow\boxed{4.667\le v_x\le5.368\ \mathrm{m/s}}.$$

(b) Result and interpretation

The extremizing pairs are $$(p_x,m)_{\min}=(9.8,2.1),\qquad(p_x,m)_{\max}=(10.2,1.9).$$ Pairing lower with lower and upper with upper excludes these allowed combinations and understates the interval.

Physical check

Both endpoints are attainable because the input bounds are independent.