Momentum and Its Conservation
Momentum quantifies an object's motion and is defined by its mass and velocity. In any isolated interaction, like a collision or explosion, the total momentum of the system is always conserved, a key principle for solving problems in one dimension.
Part of the ESAT Physics syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Momentum is a vector. Direction is crucial; always define a positive direction and use negative signs for motion in the opposite direction.
- The Law of Conservation of Momentum states that for a closed system (no external forces), the total momentum before an event equals the total momentum after.
- This conservation law applies to all interactions, including collisions where objects bounce apart and inelastic collisions where they stick together.
- A resultant force is required to change an object's momentum. The magnitude of this force is equal to the rate at which the momentum changes.
- The units of momentum are kg m/s, which is dimensionally equivalent to the Newton-second (Ns).
Diagram
› Why does this happen?
Why is momentum conserved in a collision?
This is a direct consequence of Newton's Third Law, but it only applies to a 'closed system' where no external forces (like friction or air resistance) are acting. Imagine two snooker balls colliding. When they hit, one ball pushes on the other with a force. According to Newton's Third Law, the second ball pushes back on the first with a force that is exactly equal in size and opposite in direction. The two balls are in contact for the exact same amount of time. Since change in momentum (also called impulse) is Force × time, the impulse on each ball is equal and opposite. This means whatever momentum one ball loses, the other ball gains that exact same amount. Because the momentum changes cancel each other out, the total momentum of the two-ball system stays constant.
What's the link between force and momentum?
The link comes from Newton's Second Law, which states that a resultant force causes acceleration: F = m a. We also know that acceleration is the change in velocity over time: a = (v - u) / t. By substituting the formula for acceleration into Newton's Second Law, we get: F = m × (v - u) / t. If we multiply out the bracket, this becomes F = (mv - mu) / t. We know that momentum (p) is mass × velocity, so 'mv' is the final momentum and 'mu' is the initial momentum. Therefore, 'mv - mu' is the change in momentum. This gives us the very important relationship: F = (change in momentum) / t. This shows that the resultant force on an object is equal to the rate of change of its momentum.
Formulae
p = m v To find the momentum of a single object with mass 'm' moving at velocity 'v'.
Σpinitial = Σpfinal For any collision or explosion in an isolated system. This expands to m1*u1 + m2*u2 = m1*v1 + m2*v2 for two-body interactions.
F = Δp / Δt = (mv - mu) / t To calculate the average resultant force 'F' that causes a change in momentum 'Δp' over a time interval 'Δt'.
Definitions
- Momentum (p)
- A vector quantity that is the product of an object's mass and its velocity. It measures the 'quantity of motion' an object possesses.
- Isolated System
- A collection of interacting objects on which no net external force acts. Momentum is only conserved within such a system.
- Impulse
- The change in momentum of an object (Δp). It is equal to the product of the resultant force acting on the object and the time for which it acts (FΔt).
Worked example
A 0.5 kg ball is travelling horizontally at 6 m/s to the right. It strikes a vertical wall and rebounds at 4 m/s to the left. If the ball is in contact with the wall for 0.02 s, what is the magnitude of the average force exerted by the wall on the ball?
- 1
Define 'to the right' as the positive direction.
This means the initial velocity, u = +6 m/s, and the final velocity, v = -4 m/s.
- 2
Calculate the initial momentum:
pinitial = m × u = 0.5 kg × 6 m/s = 3 kg m/s - 3
Calculate the final momentum:
pfinal = m × v = 0.5 kg × (-4 m/s) = -2 kg m/s - 4
Calculate the change in momentum (impulse):
Δp = pfinal - pinitial = (-2) - (3) = -5 kg m/s - 5
Use the force-momentum relationship:
F = Δp / Δt - 6
Substitute the values:
F = (-5 kg m/s) / 0.02 s - 7
Calculate the result:
F = -5 / (2/100) = -5 × (100/2) = -5 × 50 = -250 NThe magnitude is 250 N.
Answer: 250 N
Common mistakes
- ×Forgetting the vector nature of momentum. A velocity in the opposite direction must have a negative sign. This is the single biggest cause of incorrect answers.
- ×Making arithmetic errors, especially when adding or subtracting negative numbers during momentum conservation calculations.
- ×Confusing momentum (mv) with kinetic energy (0.5mv2). Remember that momentum is always conserved in a closed system, but kinetic energy is only conserved in perfectly elastic collisions.
- ×Calculating the change in momentum as 'final speed - initial speed' instead of 'final momentum - initial momentum'. For rebounds, this ignores the crucial sign change and gives a smaller, incorrect value for Δp.
No-calculator tips
- ✓To divide by decimals like 0.02, convert them to fractions first (e.g., 0.02 = 2/100 = 1/50). Dividing by a fraction is the same as multiplying by its reciprocal (e.g., dividing by 1/50 is the same as multiplying by 50).
- ✓In recoil or explosion problems starting from rest, the initial momentum is zero. Therefore, the final momenta of the parts must be equal in magnitude and opposite in direction. The ratio of their speeds will be the inverse of the ratio of their masses.
- ✓Before calculating, quickly estimate the outcome. If a heavy object hits a light stationary object, you expect the heavy object to slow down a little and the light object to move off quickly. If your answer contradicts this intuition, re-check your signs and arithmetic.