1.4.1 · D5Momentum & Collisions
Question bank — Linear momentum p = mv
The one symbol this whole page rests on
Before any trap, let's pin down the star of the show right here so nothing is borrowed on trust.
Two words we lean on throughout — see them side by side in the picture below:
- A vector = size and direction (an arrow). It can be negative in 1D or split into / parts in 2D.
- A scalar = size only, no direction (a plain number like kilograms or joules).
Look at the figure: the same object gets a longer arrow when is longer, and a flipped arrow when reverses — but the thickness (mass) never changes sign. That single picture is behind roughly half the traps on this page.
True or false — justify
TRUE or FALSE: "A momentum answer is complete once you state its size, e.g. — the direction is optional."
FALSE. Momentum is a vector, so its direction is part of the answer; a bare number leaves it half-finished (compare the two arrows in the figure above).
A heavy truck at rest has large momentum because it has large mass.
False. ; if then no matter how large is. Mass alone is not momentum — it must be moving.
Two objects with the same speed always have the same momentum.
False. Same speed is not same velocity (direction may differ) and not same mass. Momentum needs matching mass and matching velocity vector to be equal.
If two carts have equal momentum, they must have equal kinetic energy.
False. , so for the same the lighter cart has more . Equal momentum with different masses gives different energies (see the -vs- curve later).
Doubling an object's speed doubles both its momentum and its kinetic energy.
False for energy. doubles (linear in ), but quadruples (goes as ). Only momentum doubles.
In one dimension, momentum can be a negative number.
True. In 1D velocity carries a sign for direction, so is negative for leftward motion — the minus sign is physical, not an error.
Mass can be negative in the formula .
False. Mass is always a positive scalar; the sign of momentum comes entirely from the velocity's direction, never from the mass.
and are two entirely different laws.
False. is the general law; is the special case obtained when mass is constant, by pulling out of the derivative.
The SI unit of momentum, , is the same as the unit of impulse.
True. Impulse is , and — impulse and momentum share units because impulse is a change in momentum (see the force–time area figure below and Impulse–Momentum Theorem).
Momentum is conserved in every collision, but kinetic energy is not.
True. Momentum conserves whenever no external force acts (Newton's third law cancels the internal pushes); conserves only in elastic collisions.
The algebra links, shown step by step
Two of the traps below turn on formula-links that students often memorise rather than see. Here is where they come from, one honest step at a time.
Spot the error
"The ball's momentum is joules." — find the mistake.
The unit is wrong: joules measure energy (a scalar). Momentum is in and needs a direction. The student confused with .
"Cart A () meets cart B (); total momentum ."
Signs were dropped. As a vector sum it is ; opposite-direction arrows partially cancel (see the head-to-tail picture below), they don't add.
"A rocket burns fuel, so I'll use with the current mass."
Wrong law. The rocket's mass changes as fuel leaves, so can't be pulled out of the derivative — you must use , which keeps the changing mass inside.
"Both pucks have , so their velocities are equal."
Magnitude isn't the whole vector. Equal says the arrows are the same length, not the same direction or the same mass — many different give the same magnitude.
"To get total momentum in 2D, I add the magnitudes: ."
You can't add magnitudes. Add components separately, and , then take . Magnitudes only add when the vectors point the same way — the parallelogram figure below shows why but the true sum is .
"Kinetic energy equals , since and ."
Off by a factor. (proof just above). The correct link is .
Why questions
Why is the combination special, and not or ?
Because Newton's law states — momentum is exactly the quantity whose rate of change equals force. The force–time area figure below shows this literally: the area under a force curve is the momentum handed over, which no other combination of and gives.
Why can momentum be conserved even while kinetic energy is lost?
Internal forces come in equal-opposite pairs (Newton's Third Law), so they cancel in the vector total (equal arrows pointing opposite ways sum to zero) regardless of any energy that turns into heat, sound, or deformation. Energy has no such cancellation guarantee.
Why does treating momentum as a vector make collisions easier?
The signs (1D) or components (2D) do the cancelling and adding automatically, so "before = after" becomes clean arithmetic on numbers with signs instead of case-by-case reasoning about direction.
Why does a fast light object and a slow heavy object hurt differently even with equal momentum?
Equal momentum means equal "stopping difficulty," but the light-fast one carries more kinetic energy ( — see the steep small- end of the curve), so it can do more work (more damage) as it stops.
Why is total momentum written for a whole system?
Summing factors into total mass times the centre-of-mass velocity, so the messy multi-particle motion collapses to a single point moving with the total momentum.
Why does still work perfectly for a car, but fail for a rocket?
A car's mass is essentially constant, so leaves the derivative cleanly. A rocket sheds mass every second, so the part can't be ignored — only captures it.
Edge cases
An object sits perfectly still. What is its momentum, and its direction?
. Zero momentum has no direction — it's the one case where the "vector" collapses to nothing.
A ball moving right at and an identical ball moving left at : total momentum?
. The system's total momentum is zero even though each ball is clearly moving — the arrows exactly cancel.
Can a system have zero total momentum but nonzero total kinetic energy?
Yes. The two opposite-moving balls above sum to , yet each contributes positive , so the energy total is strictly positive.
What happens to as an object's mass shrinks toward zero at fixed momentum?
blows up toward infinity — a lighter object at the same momentum must move ever faster, and dominates. The -vs- curve (figure s02) shoots up on its left edge — this is exactly that behaviour.
A single free particle with no forces on it: does its momentum change over time?
No. With , , so stays constant — this is Newton's first law hiding inside the momentum form (see Conservation of Linear Momentum).
Two particles move so their centre of mass is at rest. What is the system's total momentum?
Exactly zero, since total momentum and . Each particle may move, but their momenta are equal-and-opposite.
If mass is truly constant, is there any physical difference between using and ?
No difference in the answer — they give identical results. The distinction only appears when changes; for constant mass they are literally the same equation rearranged.
Connections
- Impulse–Momentum Theorem — impulse is the area-under-force-time (figure s04) and equals the change in momentum.
- Conservation of Linear Momentum — the "no external force" edge cases here.
- Newton's Second Law — the vs traps.
- Newton's Third Law — why internal collision forces cancel.
- Elastic and Inelastic Collisions — momentum always, energy only sometimes.
- Kinetic Energy — the contrast at the heart of the traps.
- Centre of Mass — total momentum .