3.1 Momentum and Newton’s laws of motion
- Syllabus
- 9702–2028–2029
- Topic
- 3.1
- Level
- AS
Learning objectives
3.1.1Mass is the property of an object that resists change in motion• understand that mass is the property of an object that resists change in motion3.1.2F = ma and solve problems using it, understanding that acceleration and• recall F = ma and solve problems using it, understanding that acceleration and resultant force are always in the same direction3.1.3Linear momentum as the product of mass and velocity• define and use linear momentum as the product of mass and velocity3.1.4Force as rate of change of momentum• define and use force as rate of change of momentum3.1.5Each of Newton’s laws of motion• state and apply each of Newton’s laws of motion3.1.6The concept of weight as the effect of a gravitational field on a mass and• describe and use the concept of weight as the effect of a gravitational field on a mass and recall that the weight of an object is equal to the product of its mass and the acceleration of free fall
Mass measures resistance to acceleration, not the amount of gravitational pull
Mass is inertia: for a given resultant force, a larger mass produces a smaller acceleration through F=ma. It is a scalar measured in kilograms.
Keep mass separate from weight, which is mg, and use the chosen body’s mass in its equation of motion.
The same 10 N force gives 2 m s⁻² to a 5 kg object but 1 m s⁻² to a 10 kg object.
Mass does not depend on local g; an object’s weight can change while its inertia remains the same.
Resultant force and acceleration point in the same direction
Fresultant=ma
In F = ma, F is the vector sum of all forces acting on one chosen body. Its acceleration is in the same direction as that resultant force; for fixed mass, a larger resultant produces a proportionally larger acceleration.
Choose a positive direction, draw only forces acting on the body, add their signed components, then divide the resultant by mass.
A 850 kg car has 1600 N forwards and 1200 N backwards. Resultant force = 400 N forwards, so a = 400/850 = 0.47 m s⁻² forwards.
Do not substitute one applied force when another force opposes it. Zero resultant force means zero acceleration, not necessarily zero velocity.
Linear momentum is mass multiplied by velocity
p=mv
Linear momentum p is a vector in the same direction as velocity v. Mass m is scalar, so reversing velocity reverses momentum. The SI unit is kg m s⁻¹, equivalent to N s.
Take right as positive. A 0.10 kg ball changes velocity from +20 m s⁻¹ to −15 m s⁻¹: initial p = +2.0 kg m s⁻¹, final p = −1.5 kg m s⁻¹, and change Δp = −3.5 kg m s⁻¹.
Use velocity, not speed: momentum change is final momentum minus initial momentum. Momentum conservation in interactions is taught separately in Topic 3.3.
Force is the rate at which momentum changes
Fresultant=ΔtΔp=Δtpfinal−pinitial
The average resultant force has the direction of the momentum change. A larger momentum change in the same time, or the same change in less time, requires a larger average force.
A 0.20 kg ball moving at +14.0 m s⁻¹ rebounds at −7.0 m s⁻¹ in 0.60 s. Δp = 0.20(−7.0 − 14.0) = −4.2 kg m s⁻¹, so average F = −4.2/0.60 = −7.0 N: 7.0 N opposite to its initial motion.
Force is change in momentum divided by the time for that change, not momentum divided by time. For constant mass this definition gives F = ma.
Newton's three laws connect motion, resultant force and interactions
| Law | Statement | How to apply it |
|---|---|---|
| First | An object remains at rest or moves with constant velocity unless a resultant force acts. | Zero resultant means zero acceleration. |
| Second | Resultant force equals rate of change of momentum; for constant mass, F = ma. | Add forces on one body, then calculate its acceleration. |
| Third | When A exerts a force on B, B simultaneously exerts an equal-magnitude, opposite-direction force of the same type on A. | Name both different bodies in the pair. |
For a book resting on a table, weight and normal contact force balance on the book: this applies the first/second-law resultant. The third-law partner of the table's force on the book is the book's force on the table; the partner of Earth's gravitational force on the book is the book's gravitational force on Earth.
Third-law forces never cancel in one object's free-body diagram because they act on different objects. Balanced forces act on the same object and need not be the same type.
Weight is the gravitational force on a mass
W=mg
Weight W is the force exerted by a gravitational field on mass m. It acts in the direction of the gravitational field. Gravitational field strength g has unit N kg⁻¹, numerically equivalent to acceleration of free fall in m s⁻².
| Property | Mass | Weight |
|---|---|---|
| Meaning | resistance to change in motion | gravitational force on the mass |
| Type | scalar | vector |
| SI unit | kg | N |
| Change with local g? | no | yes |
A 5.0 kg object has weight 5.0 × 9.8 = 49 N near Earth. Where g = 1.6 N kg⁻¹, its mass remains 5.0 kg but its weight is 8.0 N.
A balance may be calibrated to display mass, but weight itself is measured in newtons. An object's velocity does not determine its weight.