Section 7

Linear momentum, impulse and collisions

viz

\vec p=m\vec v

Linear momentum is the product of mass and velocity.

View detail →

Second law in momentum form

viz

\sum\vec F=\frac{d\vec p}{dt}

For constant mass

\sum\vec F=m\vec a

The net force is the derivative of linear momentum with respect to time.

View detail →

Impulse

viz

\vec J=\int_{t_i}^{t_f}\vec F\,dt

For constant force

\vec J=\vec F\,\Delta t

Impulse is the integral of force with respect to time.

View detail →

Impulse-momentum theorem

viz

\vec J=\Delta\vec p =\vec p_f-\vec p_i

Impulse equals the change in linear momentum.

View detail →

Conservation of linear momentum

viz

\sum\vec p_i=\sum\vec p_f

The total initial linear momentum equals the total final linear momentum.

negligible net external impulse.

View detail →

Perfectly inelastic 1D collision

viz

m_1v_{1i}+m_2v_{2i} =(m_1+m_2)v_f

the objects remain joined together after the collision.

View detail →

Elastic collision

viz

Momentum is conserved

m_1v_{1i}+m_2v_{2i} =m_1v_{1f}+m_2v_{2f}

and the kinetic energy

\frac12m_1v_{1i}^2+\frac12m_2v_{2i}^2 = \frac12m_1v_{1f}^2+\frac12m_2v_{2f}^2

In an elastic collision, linear momentum and kinetic energy are conserved.

View detail →

Center of mass

viz

\vec r_{\text{CM}} = \frac{\sum_i m_i\vec r_i}{\sum_i m_i}

For a continuous distribution

\vec r_{\text{CM}}=\frac{1}{M}\int\vec r\,dm

The center of mass is the mass-weighted average of the positions.

View detail →

Center-of-mass velocity

viz

\vec v_{\text{CM}}= \frac{\sum_i m_i\vec v_i}{M} = \frac{\vec P_{\text{total}}}{M}

The center-of-mass velocity is the total linear momentum divided by the total mass.

View detail →

Section 7

Linear momentum, impulse and collisions

Linear momentum

viz

\vec p=m\vec v

Linear momentum is the product of mass and velocity.

View detail →

Second law in momentum form

viz

\sum\vec F=\frac{d\vec p}{dt}

For constant mass

\sum\vec F=m\vec a

The net force is the derivative of linear momentum with respect to time.

View detail →

Impulse

viz

\vec J=\int_{t_i}^{t_f}\vec F\,dt

For constant force

\vec J=\vec F\,\Delta t

Impulse is the integral of force with respect to time.

View detail →

Impulse-momentum theorem

viz

\vec J=\Delta\vec p =\vec p_f-\vec p_i

Impulse equals the change in linear momentum.

View detail →

Conservation of linear momentum

viz

\sum\vec p_i=\sum\vec p_f

The total initial linear momentum equals the total final linear momentum.

negligible net external impulse.

View detail →

Perfectly inelastic 1D collision

viz

m_1v_{1i}+m_2v_{2i} =(m_1+m_2)v_f

the objects remain joined together after the collision.

View detail →

Elastic collision

viz

Momentum is conserved

m_1v_{1i}+m_2v_{2i} =m_1v_{1f}+m_2v_{2f}

and the kinetic energy

\frac12m_1v_{1i}^2+\frac12m_2v_{2i}^2 = \frac12m_1v_{1f}^2+\frac12m_2v_{2f}^2

In an elastic collision, linear momentum and kinetic energy are conserved.

View detail →

Center of mass

viz

\vec r_{\text{CM}} = \frac{\sum_i m_i\vec r_i}{\sum_i m_i}

For a continuous distribution

\vec r_{\text{CM}}=\frac{1}{M}\int\vec r\,dm

The center of mass is the mass-weighted average of the positions.

View detail →

Center-of-mass velocity

viz

\vec v_{\text{CM}}= \frac{\sum_i m_i\vec v_i}{M} = \frac{\vec P_{\text{total}}}{M}

The center-of-mass velocity is the total linear momentum divided by the total mass.

View detail →