R = (sum of m_i r_i) / (sum of m_i)
The centre of mass position equals the sum of mass times position, divided by the total mass.
Centre of mass position for a discrete set of particles.
- R
- position vector of the centre of mass (m)
- m_i
- mass of the i-th particle (kg)
- r_i
- position vector of the i-th particle (m)
Use when — The system is a finite collection of point masses with known positions.
Common trap — Dividing by the number of particles instead of the total mass.
R = (1/M) * integral(r dm)
The centre of mass position equals the integral of position times the mass element, divided by total mass.
Centre of mass position for a continuous mass distribution.
- M
- total mass of the body (kg)
- dm
- infinitesimal mass element (kg)
Use when — The mass is spread continuously along a length, over a surface, or through a volume.
Common trap — Forgetting to express dm in terms of the same variable used for the integration limits.
V_cm = (sum of m_i v_i) / M
Centre of mass velocity equals the sum of mass times velocity, divided by total mass.
Velocity of the centre of mass in terms of the individual particle velocities.
- V_cm
- velocity of the centre of mass (m/s)
- v_i
- velocity of the i-th particle (m/s)
Use when — Individual particle velocities are known and the centre of mass velocity is required.
Common trap — Using an unweighted average of the velocities.
F_ext = M a_cm
Net external force equals total mass times the acceleration of the centre of mass.
The centre of mass accelerates according to the net external force on the system.
- F_ext
- net external force on the system (N)
- a_cm
- acceleration of the centre of mass (m/s^2)
Use when — You need the motion of the centre of mass rather than the motion of individual parts.
Common trap — Including internal forces, such as those from an explosion, in F_ext.
p = m v
Momentum equals mass times velocity.
Linear momentum of a particle or, summed, of a system.
- p
- linear momentum (kg m/s)
Use when — Setting up a momentum-based equation for a particle or a system of particles.
Common trap — Treating momentum as a scalar and adding magnitudes without regard to direction.
sum of p_before = sum of p_after, when F_ext = 0
Total momentum before an interaction equals total momentum after it, when net external force is zero.
Total linear momentum of an isolated system is conserved.
- p_before
- momentum of a part of the system before the interaction (kg m/s)
- p_after
- momentum of a part of the system after the interaction (kg m/s)
Use when — The net external force on the chosen system is zero, or negligible, over the interval considered.
Common trap — Applying the equation along only one axis when the motion is two-dimensional.
J = integral(F dt) = change in p
Impulse equals the time integral of force and equals the resulting change in momentum.
Impulse equals the change in momentum it produces.
- J
- impulse (N s)
- F
- applied force (N)
Use when — A force acts for a known or bounded time interval and the resulting momentum change is required.
Common trap — Using average force without confirming the time interval used matches the interaction duration.
v1' = ((m1 - m2)/(m1 + m2)) u1 + (2 m2/(m1 + m2)) u2
The final velocity of body one is a weighted combination of both initial velocities, using the mass ratio.
Final velocity of body 1 after a one-dimensional elastic collision with body 2.
- u1, u2
- initial velocities of bodies 1 and 2 (m/s)
- v1'
- final velocity of body 1 (m/s)
- m1, m2
- masses of bodies 1 and 2 (kg)
Use when — The collision is one-dimensional and both momentum and kinetic energy are conserved.
Common trap — Using this result for an inelastic collision, where kinetic energy is not conserved.
v_common = (m1 u1 + m2 u2) / (m1 + m2)
The common final velocity equals total momentum before collision divided by total mass.
Common final velocity of two bodies that stick together after collision.
- v_common
- common final velocity (m/s)
Use when — The collision is perfectly inelastic and the bodies move together afterward.
Common trap — Also assuming kinetic energy is conserved; it is not, in a perfectly inelastic collision.
e = (speed of separation along line of impact) / (speed of approach along line of impact)
The coefficient of restitution equals the relative speed of separation divided by the relative speed of approach, along the line of impact.
The coefficient of restitution compares relative speed after and before collision, along the line of impact.
- e
- coefficient of restitution (dimensionless)
Use when — The collision is one-dimensional, or the components along the line of impact of an oblique collision are being analysed.
Common trap — Applying e to the tangential component of velocity instead of the line-of-impact component.
p1_cm = -p2_cm
In the centre of mass frame, the momentum of body one is equal in magnitude and opposite in direction to the momentum of body two.
In the centre of mass frame of a two-body system, the two momenta are equal and opposite.
- p1_cm, p2_cm
- momenta of bodies 1 and 2 in the centre of mass frame (kg m/s)
Use when — Analysing a two-body interaction is simpler from the centre of mass frame than from a fixed lab frame.
Common trap — Forgetting to transform the answer back to the lab frame if the question asks for lab-frame quantities.
mu = (m1 m2) / (m1 + m2)
Reduced mass equals the product of the two masses divided by their sum.
Reduced mass of a two-body system, used to rewrite relative motion as an equivalent one-body problem.
- mu
- reduced mass (kg)
Use when — A two-body interaction is reformulated in terms of the relative coordinate between the two bodies.
Common trap — Using reduced mass in place of total mass in the centre of mass motion equation; they serve different purposes.
