Defend calculus and conservation across a changing system

Write the boundaries before selecting a law

A virtual 2 kg cart begins at x=v=0. A prescribed net force F(t)=6t−3t² acts until selected tau, with the coefficients carrying the SI units required for force. A resting captured mass is positioned at the calculated pulse-end location. The joined pair then oscillates on an ideal k=8 N/m spring, initially unstretched. These are three different intervals: forced translation, short sticking capture and free oscillation.

Integrate rather than freeze the endpoint force

During the pulse, p(t)=integral F dt=3t²−t³, v=p/2 and x=t³/2−t⁴/8. The force at the final instant is not a constant force over the earlier path. At tau=2, final force happens to be zero, but the integrated momentum is 4 kg m/s. The cart has traveled 2 m and moves at 2 m/s. Zero endpoint acceleration does not erase the preceding impulse.

Change the system for the collision

Select both masses during capture and stipulate negligible external impulse over its short duration. If the captured mass is b, common velocity is p/(2+b). Kinetic energy before capture is p²/4 and afterward p²/[2(2+b)]. Their difference becomes internal energy in the sticking interaction. Momentum conservation across this interval does not justify conserving kinetic energy or treating the earlier external pulse as internal.

Carry the new initial conditions into the oscillator

At capture time reset the local time to u=0 and measure spring displacement from its equilibrium. The joined mass M=2+b begins at x=0 with velocity v_join, so omega=sqrt(8/M), x(u)=(v_join/omega)sin(omega u), and amplitude A=v_join/omega. Spring and joined-body mechanical energy is conserved afterward under the stated ideal conditions. Using the pre-capture energy would produce an excessive amplitude.

Retain an unfamiliar derivation and a critique

Before viewing feedback, write a changed-pulse or changed-mass prediction in the retained course workspace. Include the integrated functions, each system/time boundary, a dimension check and one independently recalculated endpoint. Record which answer or worked solution you viewed. Then critique a tempting incorrect approach and preserve a revision rather than replacing the initial attempt. This authored workflow is not a secure examination or evidence that a learner has performed a laboratory.

Worked original case

At tau=2 and captured mass2 kg, momentum is4, common speed1 m/s, and K changes4→2 J during capture. The final oscillator has omega=sqrt2 and A=1/sqrt2 m. The first turning point occurs pi/(2sqrt2) seconds after capture, not after the initial pulse begins.

Investigation

Compare tau=1 and2 at captured mass2, then change the captured mass. Retain pulse integrals, capture energy loss and post-capture phase for each. Explain why the station is placed at x(tau) in each virtual scenario rather than treating the changing endpoint as the same fixed apparatus. Complete a new written force-function case without using the generator formula as independent evidence.

Public numerical transfer practice

Try a new practice case

For tau=1 s and captured mass2 kg in the taught model, what is kinetic energy immediately after capture in joules?

Numerical result in the requested units Check this practice result

This public, unsaved check is not a secure assessment. Explain the derivative, integral or boundary in the retained course evidence workspace.

Sources and evidence boundary

Original derivations and assigned numerical cases checked 30 October 2026 against the official AP Physics C Mechanics framework and OpenStax mechanics and official laboratory guidance. These are supplied or generated records, not physical measurements or performed laboratory time. Do not construct rotating equipment, release loads or conduct launch experiments. Public feedback and retained self-reports do not certify independent mastery.