1.6 Momentum

Learning objectives

1.6.1Momentum as mass × velocity• Define momentum as mass × velocity; recall/use: p = mv 1.6.2Impulse as force × time for which• Define impulse as force × time for which force acts; recall/use: impulse = F∆t = ∆(mv)1.6.3Apply the principle of the• Apply the principle of the conservation of momentum to solve simple problems in one dimension 1.6.4Resultant force as change in momentum• Define resultant force as change in momentum per unit time; recall/use: F = Δp/Δt

Momentum is the product of an object's mass and velocity.

$p=mv$

Symbol	Meaning	SI unit
$p$	momentum	kg m/s, equivalent to N s
$m$	mass	kg
$v$	velocity	m/s

Momentum is a vector and points in the same direction as velocity. In one dimension, choose one direction as positive and give motion in the opposite direction a negative velocity and momentum.

For constant mass, calculate change in momentum as final minus initial: $\Delta p=mv-mu=m(v-u)$ . Keep the velocity signs when an object reverses direction.

Do not use speed alone when direction changes. An object rebounding at the same speed has the same momentum magnitude but the opposite momentum, so its momentum has changed.

Impulse is the force multiplied by the time for which the force acts, and it equals the change in momentum.

$I=F\Delta t=\Delta p=mv-mu$

Step	Action
1	choose a positive direction and assign signs to the initial and final velocities
2	calculate $\Delta p=mv-mu$
3	set impulse $I=\Delta p$
4	use $I=F\Delta t$ to find force or contact time if required

Impulse is measured in newton seconds (N s), which is equivalent to kg m/s. Convert milliseconds to seconds before using $F\Delta t$ .

For the same change in momentum, increasing the collision or stopping time reduces the average force. This is why crumple zones, padding and moving the hands backwards while catching reduce injury.

If an object rebounds, final and initial velocities have opposite signs: use final minus initial rather than subtracting the two speeds as unsigned numbers.

In an isolated system with no resultant external force, total momentum remains constant.

$\sum p_{before}=\sum p_{after}$

Step	Action
1	define one direction as positive
2	write every initial momentum $mv$ , including zero for a stationary object
3	write every final momentum with the same sign convention
4	equate the signed totals and solve for the unknown velocity
5	interpret a negative answer as motion opposite to the chosen positive direction

If two objects stick together, their final masses combine: $m_1u_1+m_2u_2=(m_1+m_2)v$ . For recoil or separation from rest, the two final momenta are equal in magnitude and opposite in direction.

Apply the principle only to the stated one-dimensional system during the event. External forces such as friction must be negligible over the collision or explosion time.

Momentum is conserved in an isolated collision, but kinetic energy need not be. Never add opposite-direction momenta as positive magnitudes.

The resultant force on an object equals its change in momentum per unit time.

$F=\frac{\Delta p}{\Delta t}=\frac{mv-mu}{\Delta t}$

Step	Action
1	choose a positive direction
2	calculate the signed change $\Delta p=p_{final}-p_{initial}$
3	convert the time interval to seconds
4	divide by $\Delta t$ and state the force direction

A larger momentum change in the same time gives a larger average resultant force. For the same momentum change, a longer time gives a smaller average resultant force.

Rearranging gives $F\Delta t=\Delta p$ , so the area represented by force × time is the impulse. For constant mass, this result is consistent with $F=ma$ .

Use change in momentum, not momentum alone. $F=p/\Delta t$ is valid only when the initial momentum is zero or when $p$ explicitly means the momentum change.

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—1.7.1 EnergyRead

1.6 Momentum

Learning objectives

Calculate momentum and momentum change

Connect impulse to change in momentum

Conserve momentum in one dimension

Calculate force from momentum change

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