238 Momentum
A measure of an object’s motion that is used to describe how hard it is to stop or redirect that object.
\(\mathbf{p}=m\mathbf{v}\). Linear momentum is the product of mass and velocity. It is a vector in the direction of the velocity. This principle is used to assign a single motion quantity to a body.
The linear momentum is
\[ \mathbf{p} = m\mathbf{v} \]
where
- \(\mathbf{p}\) is the momentum.
- \(m\) is the mass.
- \(\mathbf{v}\) is the velocity.
The impulse-momentum theorem. A force delivers an impulse equal to the change of momentum. This principle is used to compute a velocity change from a known force history.
The impulse-momentum theorem is
\[ \mathbf{J} = \displaystyle\int\mathbf{F}\,dt = \Delta\mathbf{p} \]
where
- \(\mathbf{J}\) is the impulse.
- \(\mathbf{F}\) is the force.
- \(t\) is time.
- \(\mathbf{p}\) is the momentum.
Conservation in an isolated system. The total momentum of an isolated system is constant. An isolated system feels no net external force. This principle is used to analyze collisions.
Conservation of momentum is
\[ \mathbf{F}_{\mathrm{ext}} = 0 \implies \sum_{i}\mathbf{p}_{i} = \mathrm{constant} \]
where
- \(\mathbf{F}_{\mathrm{ext}}\) is the net external force.
- \(\mathbf{p}_{i}\) is the momentum of particle \(i\).
Translation symmetry. Momentum is conserved because the laws of physics do not depend on where an experiment is done. This principle is used to identify momentum with spatial-translation symmetry.
The quantum momentum operator. In quantum mechanics momentum is represented by the operator \(-i\hbar\nabla\). This principle is used to compute \(\langle p\rangle\) from a wavefunction.
The momentum operator is
\[ \hat{\mathbf{p}} = -i\hbar\nabla \]
where
- \(\hat{\mathbf{p}}\) is the momentum operator.
- \(\hbar\) is the reduced Planck constant.
- \(\nabla\) is the gradient.
Note: Heavier objects and faster objects both carry more momentum. Electromagnetic fields and light also carry momentum. Energy and momentum combine into a relativistic four-vector.
238.1 References
- Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — mass times velocity; impulse.
- Jaffe, R. L., & Taylor, W. The Physics of Energy. Cambridge University Press, 2018. — conservation; kinetic energy link.
- Susskind, L., & Friedman, A. Special Relativity and Classical Field Theory: The Theoretical Minimum. Basic Books, 2017. — four-momentum; quantum operator.
- Feynman, R. P., Leighton, R. B., & Sands, M. The Feynman Lectures on Physics. — field and light momentum; wave oscillations.
- Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — conservation from translation invariance.