Force and Motion (I): Newton's First and Second Laws
AdvancedNewton's First Law: with no net force an object stays at rest or in uniform straight-line motion, and mass measures inertia; Newton's Second Law F = ma links net force, mass and acceleration — with free-body diagrams, the core of mechanics problem-solving.
- Newton's First Law: with no net force, stay at rest or in uniform straight-line motion — force changes motion
- Inertia = resistance to a change of state of motion; the larger the mass, the greater the inertia
- Newton's Second Law F = ma: acceleration ∝ net force, ∝ 1/mass; 1 N = 1 kg m s⁻²
文體小知識：科學探究記敘——由伽利略斜面實驗說到驗證 F = ma 的實驗探究，記述牛頓第一及第二定律如何從觀察與實驗歸納出來。
§Text Passage
1A force is a push or pull that changes an object's state of motion or its shape, measured in newtons (N). Ancient thinkers believed motion needed a force to maintain it, but Galileo's inclined-plane experiments argued otherwise: a ball rolling down one slope rises almost to its original height on the opposite slope, and if the opposite slope were made horizontal, the ball would roll on forever. Newton summed this up as the First Law: an object with no net force (or zero net force) stays at rest or continues in uniform straight-line motion.
2An object's resistance to any change of its state of motion is called inertia, and mass measures inertia: the larger the mass, the harder its velocity is to change. For example, when a bus brakes suddenly, passengers lurch forward because their bodies try to keep their original speed while the bus slows; seat belts provide a restraining force in a collision so passengers decelerate with the car instead of flying forward. Inertia also explains why shaking a tree brings down its fruit — the fruit keeps still while the branch moves and the stem snaps.
3Newton's Second Law states: an object's acceleration is proportional to the net force and inversely proportional to its mass — net force equals mass times acceleration. The net force is the combined effect of all forces acting, measured in newtons; one newton gives a 1 kg mass an acceleration of 1 m s⁻². When several forces act, find the net force first. For example, pushing a 5 kg curling stone along smooth ice with a net force of 20 N gives an acceleration of 20 ÷ 5 = 4 m s⁻².
4To analyse forces, draw a free-body diagram: sketch the object as a simple shape, mark every force with an arrow, then find the net force along the direction of motion. For example, a car on a level road feels gravity, the normal support of the road, the engine's driving force and friction; gravity and support cancel, so the net force equals driving force minus friction and the car accelerates. Mass and weight must also be distinguished: mass is measured in kilograms and never changes with location; weight is the gravitational force, mass times g, measured in newtons, and it shrinks on the Moon.
5These two laws are the backbone of mechanics: the First Law says force is not needed to maintain motion but to change it, and the Second Law gives the quantitative rule of that change. With F = ma and free-body diagrams in hand, slopes, connected bodies and hanging weights all become routine; the next lesson completes the picture of the world of forces.
1Summary 重點摘要
Newton's First Law: with no net force an object stays at rest or in uniform straight-line motion, and mass measures inertia; Newton's Second Law F = ma links net force, mass and acceleration — with free-body diagrams, the core of mechanics problem-solving.
2Vocabulary cards Vocabulary
A push or pull that changes an object's state of motion or its shape; a vector measured in newtons (N).
The vector sum of all forces acting on an object (the resultant).
The unit of force: 1 N gives a 1 kg mass an acceleration of 1 m s⁻².
An object's resistance to a change of its state of motion; the larger the mass, the greater the inertia.
With no net force, an object stays at rest or in uniform straight-line motion.
The acceleration of an object is proportional to the net force and inversely proportional to its mass: F = ma.
A simplified sketch of an object with arrows showing every force acting on it, used to find the net force.
The quantity of matter in an object and the measure of its inertia, in kilograms; unchanged by location.
The gravitational force on an object, W = mg, in newtons; varies with the local gravitational field.
The force opposing relative sliding or the tendency of sliding, directed against the motion (or tendency).
3Step by step Steps
- Step 1: The concept and unit of forceA force is one object's action on another; it can change an object's state of motion — the size or direction of its velocity — or deform the object. Force is a vector with magnitude and direction, and its SI unit is the newton (N); one newton is roughly the force needed to hold up a small apple. For example, pulling open a drawer takes about 20 N, and the Earth pulls a 1 kg mass with about 9.8 N. Forces act by contact (pushing, pulling, friction) or at a distance (gravity, magnetism) — a distinction you will keep meeting.
- Step 2: Galileo's slopes and the First LawAncient thinkers believed motion needed a force to sustain it; Galileo challenged this intuition with inclined planes: a ball rolling down one slope and up another rises almost to its original height, so if the second slope is laid flat, the ball should roll on at constant speed forever. Newton wrote this as the First Law: with no net force (or zero net force) an object stays at rest or moves in uniform straight-line motion. For example, a space probe flies on for years at constant velocity after its engines shut down, because space is nearly force-free — maintaining motion needs no force; changing it does.
- Step 3: Inertia and massThe property of keeping the original state of motion and resisting change is inertia, and mass measures it: the larger the mass, the smaller the acceleration from the same force, and the harder the state of motion is to change. For example, when a bus brakes hard, passengers lurch forward out of inertia, and a loaded lorry is far harder to stop than a motorcycle; a seat belt exerts a restraining force in a crash so the passenger decelerates with the car — the physics behind mandatory seat-belt laws. Answer template: 'the object was originally …; an external force changes …; by inertia the object keeps …, therefore …'.
- Step 4: Newton's Second Law F = maNewton's Second Law: an object's acceleration is proportional to the net force and inversely proportional to its mass — F = ma. F must be the net force: combine all the forces first, then substitute; the acceleration points the same way as the net force. One newton is defined as the force giving a 1 kg mass an acceleration of 1 m s⁻². For example, a 1200 kg car under a 2400 N net force gains a = F/m = 2400 ÷ 1200 = 2 m s⁻²; the same force on a 2400 kg lorry gives only 1 m s⁻² — the bigger the mass, the harder to 'push'.
- Step 5: Free-body diagrams and the net forceA free-body diagram reduces the object to a point or simple shape and marks every force with an arrow — the standard tool for finding the net force. Draw gravity first, then contact forces (support, friction, tension), then add the forces along the direction of motion as vectors. For example, a 1500 kg car pulled forward by a 3000 N driving force against 600 N of friction has a net force of 3000 − 600 = 2400 N along the motion, so a = 2400 ÷ 1500 = 1.6 m s⁻²; vertically, gravity and support cancel and do not affect the horizontal acceleration.
- Step 6: Mass versus weightMass and weight are different concepts. Mass is the quantity of matter and the measure of inertia, in kilograms, unchanged by location; weight is the gravitational force the Earth exerts, W = mg, in newtons, varying with the local gravitational field. For example, a 60 kg astronaut weighs 60 × 9.8 = 588 N on Earth but only about 98 N on the Moon (gravity about one-sixth of the Earth's), while the mass is still 60 kg and the inertia unchanged — kicking the same object needs the same force on the Moon as on Earth.
- Step 7: Putting it together — the solution routineSecond-Law problems follow one routine: draw the free-body diagram → find the net force along the motion → substitute into F = ma for the acceleration → use the equations of motion for velocities and distances, chaining this topic to the last one. For example, a 1200 kg car under a 2400 N net force from rest for 5 s: a = 2 m s⁻², so after 5 s v = 0 + 2 × 5 = 10 m s⁻¹ and s = ½ × 2 × 5² = 25 m. Seat belts, airbags and head restraints are all applications of inertia and the Second Law.
4Diagrams Diagrams
5Worked example Example
Example: a 1200 kg car drives on a straight level road with an engine driving force of 3000 N and friction of 600 N. (a) Find the car's acceleration. (b) Find its velocity and displacement after accelerating from rest for 5 s. (c) Explain how a seat belt protects passengers in an emergency brake.
- Free-body diagram: vertically, gravity and support cancel; along the road, net force = 3000 − 600 = 2400 N forward
- Second Law: a = F ÷ m = 2400 ÷ 1200 = 2 m s⁻² forward
- Kinematics: after 5 s, v = u + at = 0 + 2 × 5 = 10 m s⁻¹; s = ½at² = ½ × 2 × 5² = 25 m
- Inertia explanation: during a sudden brake the car slows rapidly while the passengers, by inertia, keep their original speed and lurch forward; the seat belt applies a backward restraining force so they decelerate with the car instead of striking the windscreen
- Answer: (a) Acceleration 2 m s⁻² forward; (b) after 5 s the velocity is 10 m s⁻¹ and the displacement 25 m; (c) the seat belt applies a restraining force that cancels the inertia-driven lurch, letting the passengers decelerate with the car. Routine: free-body diagram → net force → F = ma → equations of motion.
6Common mistakes Mistakes
7Quick check Quick check
Q1State Newton's First Law.
Q2What is inertia, and which quantity measures it?
Q3Calculate: a net force of 2400 N acts on an 800 kg object. Find the acceleration.
Q4What is the weight of a 50 kg person on Earth and on the Moon?
Q5Why do passengers lurch forward when a bus brakes hard, and what does a seat belt do?
8Recap Recap
- Newton's First Law: with no net force, stay at rest or in uniform straight-line motion — force changes motion
- Inertia = resistance to a change of state of motion; the larger the mass, the greater the inertia
- Newton's Second Law F = ma: acceleration ∝ net force, ∝ 1/mass; 1 N = 1 kg m s⁻²
- Free-body diagram → net force → F = ma → equations of motion: the fixed routine of mechanics
- Mass (kg, constant) differs from weight (N, W = mg); seat belts and airbags apply inertia for safety