Connect force and displacement to changes in motion, follow a spring’s stored energy, and distinguish an energy transfer from its rate.
Key knowledge
For a constant-mass Newtonian particle, work by the net force along its displacement equals the change in kinetic energy K = ½mv².
Values and equations
W_net = ∫ F_net · dr = K_f − K_i; K = ½mv²
W_net
Net work along the trajectory J
F_net
Net vector force N
r
Position vector m
K
Particle kinetic energy J
m
Particle mass kg
v
Particle speed m s⁻¹
Conditions and scope
Use an inertial frame, constant mass and classical particle mechanics.
Net work includes every force on the particle; v denotes speed in the kinetic-energy expression.
Sources · 5
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
01Physics, Part I (Class XI; reprint 2026–27)National Council of Educational Research and Training (NCERT)Textbook chapter · PDFChapter Five: Work, Energy and Powerncert.nic.in
02Classical Mechanics: MIT 8.01 Course Notes (2022 notes)Peter Dourmashkin — Massachusetts Institute of Technology, MIT OpenCourseWareCourse notes · PDFChapter 13: Energy, Kinetic Energy, and Workocw.mit.edu
2026-10-10 · Particle work-energy theorem only; an extended body's internal energy and relativistic kinetic energy are not covered.
Foundational review
2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.
Sources from other places
Scope or convention differs
Classical net work equals the change in translational kinetic energy.
The theorem and braking example agree; this teaching section does not derive the full variable-force vector path integral.
Stopping power S(E)=−dE/dℓ is the mean energy loss per path length in a specified material. Collision/electronic loss involves excitation and ionization, nuclear stopping involves target recoil, and radiative loss includes bremsstrahlung where relevant. Neglecting energy-loss fluctuations, the continuous-slowing-down path range is R_CSDA(E₀)=∫₀^E₀dE/S(E); it differs from the projected depth along the initial direction when scattering deflects the particle.
The mean energy loss per actual path length defines stopping; integrating its reciprocal is a continuous-mean-loss path model, not an exact trajectory.
When external impulse is negligible, total momentum is conserved through a Newtonian collision. An elastic collision also preserves total translational kinetic energy; an inelastic collision does not. In a perfectly inelastic collision two bodies stick and have common final velocity (m₁v₁+m₂v₂)/(m₁+m₂). Their translational kinetic-energy decrease becomes other forms of energy rather than destroying total energy.
This relation concerns the stated elastic and inelastic collisions claim and its conditions; it does not assert an unqualified prerequisite or an additional theorem.
A time-independent conservative position force has path-independent work and zero work around every closed path in its domain. A potential U can be chosen so F=−∇U and U(B)−U(A)=−∫_A^B F·dr. Adding a constant to U changes neither force nor work. Local vanishing curl alone needs suitable global-domain conditions before it establishes a single-valued potential.
This relation concerns the stated conservative forces and potential energy claim and its conditions; it does not assert an unqualified prerequisite or an additional theorem.
For a Newtonian particle under time-independent conservative forces plus other forces, E_mech=K+U and ΔE_mech=W_other. When the other forces do zero work, mechanical energy stays constant. Friction can convert ordered mechanical energy into internal energy; loss of K+U is compatible with total-energy conservation for a correctly chosen combined system.
This relation concerns the stated mechanical-energy conservation and dissipation claim and its conditions; it does not assert an unqualified prerequisite or an additional theorem.
Average power is work divided by the time interval; instantaneous mechanical power is dW/dt. For a force acting at a point moving with velocity v, its power is F·v. A force perpendicular to that velocity transfers no mechanical energy through that point even if it changes the direction of motion.
This relation concerns the stated mechanical power claim and its conditions; it does not assert an unqualified prerequisite or an additional theorem.
Learning guidance · 2 / 2
Preserved learning suggestions. They are not proof, empirical evidence or mandatory prerequisites.