Momentum
AdvancedMomentum is the quantity of motion, p = mv, a vector; impulse Ft equals the change in momentum, and lengthening the collision time reduces the average force. The total momentum of a closed system is conserved — Newton's cradle and collision problems all rely on this law.
- Momentum p = mv is a vector in kg m s⁻¹; choose a positive direction first
- Impulse Ft = mv − mu, in N s; force equals the rate of change of momentum
- For a fixed momentum change, longer time means smaller average force — airbags, helmets and sand pits all lengthen the time
文體小知識：科學知識說明文——由碰撞與撞擊入手，講解動量、衝量與動量守恆定律，並用安全設計與牛頓搖籃說明守恆律的威力，是力學的收結一課。
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1Momentum describes the 'quantity of motion' of an object: it is defined as the product of mass and velocity, in kg m s⁻¹. Since velocity is a vector, so is momentum, pointing along the velocity; choose a positive direction first and give opposite directions negative signs. For example, a 0.45 kg football flying at 20 m s⁻¹ towards a goalkeeper carries momentum of 9 kg m s⁻¹; a shot put of the same speed, being far heavier, carries far more momentum and is much harder to stop.
2A force acting for a period of time changes an object's momentum. Impulse is defined as the product of force and time, in newton seconds, and equals the change in momentum: final momentum minus initial momentum. This also writes Newton's second law in another form: force equals the rate of change of momentum. For example, a racket hitting a ball with 50 N for 0.02 s gives an impulse of 1 N s, changing the ball's momentum by 1 kg m s⁻¹.
3From the impulse relation, when the change in momentum is fixed, a longer contact time means a smaller average force. Safety design applies this everywhere: airbags, the foam inside helmets and crash cushions all deliberately lengthen the impact time. For example, a long jumper landing in sand lets the grains slide, stretching the stopping time to a fraction of a second and cutting the average force greatly; on hard ground the time is tiny and the force tens of times larger, enough to break bones.
4The law of conservation of momentum states that in a closed system with no external force, the total momentum of the objects before and after a collision is unchanged. Collisions are elastic or inelastic: elastic collisions conserve kinetic energy too; inelastic ones lose part of the kinetic energy to internal and sound energy, most of all when the objects couple together — yet momentum is still conserved. The law holds because the forces between the objects are equal in size and opposite in direction.
5Newton's cradle demonstrates conservation beautifully: lift one steel ball and let it strike the rest, and one ball flies out at the far end; lift two, and two fly out — momentum and energy carried through. The fixed routine for momentum problems: choose a positive direction, write each mass and velocity with signs, set up the conservation equation, solve, then check that the direction makes sense. Rocket propulsion and the recoil of a gun are explained by the same law.
1Summary 重點摘要
Momentum is the quantity of motion, p = mv, a vector; impulse Ft equals the change in momentum, and lengthening the collision time reduces the average force. The total momentum of a closed system is conserved — Newton's cradle and collision problems all rely on this law.
2Vocabulary cards Vocabulary
The product of mass and velocity, p = mv; a vector in kg m s⁻¹.
Force times the period it acts, Ft, equal to the change in momentum mv − mu, in N s.
In a closed system with no external force, the total momentum is unchanged by a collision.
A system with no net external force; conservation of momentum holds only in this case.
An ideal collision in which kinetic energy as well as momentum is conserved.
A collision in which some kinetic energy becomes internal or sound energy, while momentum is still conserved.
How fast momentum changes with time; force equals the rate of change of momentum (another form of Newton's second law).
The backward motion of the rest of a system when mass is ejected, explained by conservation of momentum.
A row of suspended steel balls demonstrating the transfer and conservation of momentum and energy.
The constant force that would produce the same momentum change: the change in momentum divided by the time.
3Step by step Steps
- Step 1: Momentum is the 'quantity of motion'Momentum is defined as the product of mass and velocity, p = mv, in kg m s⁻¹. It is a vector along the velocity, so choose a positive direction before calculating — usually to the right or along the motion — and give velocities in the opposite direction a negative sign. For example, a 0.45 kg football flying at 20 m s⁻¹ towards a goalkeeper carries momentum 0.45 × 20 = 9 kg m s⁻¹; if the keeper parries it back at 5 m s⁻¹, the final momentum is 0.45 × (−5) = −2.25 kg m s⁻¹, and the change in momentum is −2.25 − 9 = −11.25 kg m s⁻¹, the negative sign showing the reversal.
- Step 2: Impulse: Ft = mv − muThe accumulated effect of a force acting for a time is impulse: force × time = Ft, in N s. Impulse equals the change in momentum: Ft = mv − mu, with u the initial and v the final velocity. For example, a racket striking a ball with 50 N for 0.02 s gives an impulse of 50 × 0.02 = 1 N s, changing the ball's momentum by 1 kg m s⁻¹; for a 0.1 kg ball struck back the other way, the speed change is 1 ÷ 0.1 = 10 m s⁻¹. Note that N s and kg m s⁻¹ are the same unit written two ways, since 1 N = 1 kg m s⁻².
- Step 3: Force as the rate of change of momentumRearranging Ft = mv − mu gives F = (mv − mu) ÷ t: force equals the rate of change of momentum. This is Newton's second law in another form, fully consistent with F = ma because mv − mu = m(v − u). For example, a 0.5 kg ball whose momentum grows from 2 kg m s⁻¹ to 6 kg m s⁻¹ in 0.1 s gains 4 kg m s⁻¹, so the average force is 4 ÷ 0.1 = 40 N; stretched over 0.4 s, the same change needs only 10 N. Four times the time, a quarter of the force — the inverse relation behind the safety design of the next step.
- Step 4: Longer time, smaller forceWhen the change in momentum is fixed, a longer contact time means a smaller average force — the heart of safety design. An airbag inflates during a crash, stretching the stopping time of a passenger from a few milliseconds to about 0.1 s; helmet foam, boxing gloves and gymnastics mats work the same way. For example, a 0.06 kg egg falling from a table lands at about 1.7 m s⁻¹, a momentum change of about 0.1 kg m s⁻¹; on hard flooring, stopped in 0.002 s, the average force is about 51 N and the egg breaks; on a soft cushion taking 0.05 s, it is about 2 N and the egg survives.
- Step 5: Conservation of momentum and collision sumsIn a closed system with no external force, the total momentum is the same before and after a collision: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. The routine has four steps: choose a positive direction, write the total momentum before, write the total momentum after, and solve the equation. For example, a 2 kg trolley at 3 m s⁻¹ hits a stationary 1 kg trolley and they couple together at velocity v: (2 × 3) + (1 × 0) = (2 + 1)v, so 6 = 3v and v = 2 m s⁻¹, moving to the right. Coupling is a perfectly inelastic collision: kinetic energy is lost, momentum is not.
- Step 6: Elastic and inelastic collisionsCollisions come in two kinds. In an elastic collision, momentum and kinetic energy are both conserved — collisions between molecules and atoms come close to this ideal. In an inelastic collision momentum is still conserved, but part of the kinetic energy becomes internal energy and sound; deformation, heating and noise after impact are the evidence. For example, the 2 kg and 1 kg trolleys above have kinetic energy ½ × 2 × 3² = 9 J before and ½ × 3 × 2² = 6 J after, losing 3 J to internal and sound energy. Remember: conservation of momentum almost always applies; conservation of kinetic energy only holds if the question states the collision is elastic.
- Step 7: Newton's cradle and recoilA Newton's cradle is a row of identical steel balls hung in contact: lift one ball and release it, and one ball flies out at the far end; lift two, and two fly out — momentum and energy pass through with their values intact. Recoil phenomena follow the same law: a rocket expels hot gas backwards and gains forward momentum; a gun fires a bullet forward and recoils backward. For example, a 0.01 kg bullet leaving at 400 m s⁻¹ carries momentum 4 kg m s⁻¹; a 2 kg gun then recoils at 4 ÷ 2 = 2 m s⁻¹ in the direction opposite to the bullet.
4Diagrams Diagrams
5Worked example Example
Example: a 0.45 kg football flies at 20 m s⁻¹ towards a goalkeeper, who catches and stops it in 0.1 s. (a) Find the magnitude of the change in the ball's momentum. (b) Find the average force exerted on the goalkeeper by the ball.
- (a) Initial momentum: p = mv = 0.45 × 20 = 9 kg m s⁻¹ (taking the ball's direction as positive)
- (a) After stopping, v = 0, so the change is 0 − 9 = −9 kg m s⁻¹; magnitude 9 kg m s⁻¹
- (b) From Ft = mv − mu: impulse magnitude = 9 N s
- (b) Average force F = 9 ÷ 0.1 = 90 N
- Direction check: the force on the ball opposes its motion; the force on the goalkeeper acts in the ball's original direction
- Answer: (a) Change in momentum = 0 − 9 = −9 kg m s⁻¹, magnitude 9 kg m s⁻¹ (impulse 9 N s). (b) F = Δp ÷ t = 9 ÷ 0.1 = 90 N, in the ball's original direction. Choose a positive direction first — the negative sign shows the reversal — and state the direction in words for full marks.
6Common mistakes Mistakes
7Quick check Quick check
Q1Define momentum: what are the formula and the unit?
Q2Calculate: a 0.2 kg ball flies at 5 m s⁻¹. What is its momentum?
Q3A racket hits a ball with 50 N for 0.02 s. What impulse does the ball receive?
Q4A 2 kg trolley at 3 m s⁻¹ hits a stationary 1 kg trolley and they couple. Find their common velocity.
Q5Why does an airbag reduce injuries in a crash?
8Recap Recap
- Momentum p = mv is a vector in kg m s⁻¹; choose a positive direction first
- Impulse Ft = mv − mu, in N s; force equals the rate of change of momentum
- For a fixed momentum change, longer time means smaller average force — airbags, helmets and sand pits all lengthen the time
- Closed system: momentum is conserved, m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂; inelastic collisions lose kinetic energy, never momentum
- Newton's cradle, rocket recoil and gun kick are all explained by conservation of momentum