Stem-Master
Physics / Chapter 9: Center of Mass and Linear Momentum / Section 9-3

Linear Momentum — Question 9

Section 9-3 · Physics pages 1208–1209

Question 9 Momentum cannot jump under a bounded force A sensor claims that a 1.0 kg particle's momentum jumps from +3 to −1 kg m/s at one exact instant while the net force remains bounded by 20 N in magnitude. t px 3 −1 ∆t A bounded force permits a continuous finite-duration transition, not an instantaneous momentum jump. (a) Test the claim and find the minimum time needed for that change under the force bound. (b) Could a very fast continuous transition appear as a jump in coarsely sampled data?
Show solutionHide solution
Question 9 - Solution Momentum cannot jump under a bounded force t px 3 −1 ∆t (a) Derivation and calculation The required momentum change is ∆px = −1 −3 = −4 kg m/s. |∆px| = Z Fx dt ≤ Z |Fx| dt ≤20∆t ⇒∆t ≥0.20 s . (b) Result and interpretation For a bounded force, shrinking the duration forces the impulse to zero: lim ∆t→0 |∆px| ≤lim ∆t→0 20∆t = 0. Sparse sampling can make a continuous transition look abrupt, but any resolved change shorter than 0.20 s contradicts the stated bound. Physical check The lower duration is attained by applying a constant −20 N force throughout the transition.

Question 9

Momentum cannot jump under a bounded force

A sensor claims that a 1.0kg1.0\,\mathrm{kg} particle's momentum jumps from +3+3 to −1kgm/s-1\,\mathrm{kg\,m/s} at one exact instant while the net force remains bounded by 20N20\,\mathrm N in magnitude.

Diagram shown in the original SVG worksheet.

  1. Test the claim and find the minimum time needed for that change under the force bound.

  2. Could a very fast continuous transition appear as a jump in coarsely sampled data?

Question 9 - Solution

Momentum cannot jump under a bounded force

Diagram shown in the original SVG worksheet.

(a) Derivation and calculation

The required momentum change is Δpx=−1−3=−4kgm/s.\Delta p_x=-1-3=-4\ \mathrm{kg\,m/s}. |Δpx|=|∫Fxdt|≤∫|Fx|dt≤20Δt⇒Δt≥0.20s.|\Delta p_x|=\left|\int F_x\,dt\right|\le\int|F_x|\,dt \le20\Delta t\Rightarrow\boxed{\Delta t\ge0.20\ \mathrm s}.

(b) Result and interpretation

For a bounded force, shrinking the duration forces the impulse to zero: limΔt→0|Δpx|≤limΔt→020Δt=0.\lim_{\Delta t\to0}|\Delta p_x|\le\lim_{\Delta t\to0}20\Delta t=0. Sparse sampling can make a continuous transition look abrupt, but any resolved change shorter than 0.20 s contradicts the stated bound.

Physical check

The lower duration is attained by applying a constant −20-20 N force throughout the transition.

← Section questionsChapter contents