Chapter 6 — ବଳ ଓ ଗତି ନିୟମ

In the previous chapter a body's velocity changed — a bus started, a stone fell, a car braked — and we described the change without asking its cause. The cause is force. In everyday language a force is a push or a pull. We push a door to open it, pull a bucket from a well, kick a football, stretch a rubber band, squeeze a lemon. In each case a force is applied by one body on another, and something changes.

Force cannot be seen; it is known by its effects. A force can: 1. set a body at rest in motion — a footballer kicks a stationary ball; 2. stop a moving body — a fielder catches a ball; 3. change the speed of a moving body — pedalling harder speeds a cycle, braking slows it; 4. change the direction of a moving body — a batsman deflects a ball, the Sun's pull bends the Earth's path into an orbit; 5. change the shape or size of a body — pressing a sponge, stretching a spring, flattening dough. The first four effects are changes of motion, that is, accelerations; the fifth is deformation. A force may act while bodies are in contact (a push, friction, the tension in a rope) or at a distance (gravity, magnetism, electric attraction).

Force is a vector quantity: to describe a force fully we must give its magnitude, its direction and its point of application. Pushing a door at the handle and at the hinge with the same force produce different results. The SI unit of force is the newton (N), defined later in the chapter; a force of about 1 N is the weight of a small apple, and a 1 kg mass weighs about 9.8 N.

Two facts about force run through everything that follows. First, a force always involves two bodies — the one that exerts it and the one on which it acts; a force does not exist by itself. Second, what matters for motion is the net force — the resultant of all the forces acting on a body — not any single force. A book on a table has two forces on it, its weight downward and the support of the table upward, and because they cancel the book does not move. A tug-of-war rope pulled equally from both sides stays still. This idea of forces adding up, or cancelling, leads to the distinction between balanced and unbalanced forces in the next topic.

A historical note: for two thousand years, following Aristotle, people believed that a force is needed to keep a body moving, because a cart stops when the horse stops pulling. Galileo, and then Newton, showed that this is wrong — the cart stops because of friction, another force — and that a force is needed only to change motion, not to maintain it. This reversal is the heart of the chapter.

Several forces usually act on a body at once, and the body responds to their combined effect.

Balanced forces. When the forces acting on a body are equal in magnitude and opposite in direction so that their resultant is zero, they are called balanced forces. Balanced forces do not change the state of rest or of uniform motion of a body, though they may change its shape. A wooden block on a table pulled by two equal strings in opposite directions does not move. In a tug-of-war between two equally strong teams the rope stays where it is. A book lying on a table has its weight balanced by the upward push of the table. A parachutist falling at steady speed has the downward pull of gravity balanced by the upward air resistance. In every case the net force is zero and there is no acceleration; a body under balanced forces is either at rest or moving with constant velocity.

Unbalanced forces. When the forces do not cancel — one is greater than the other, or they act in different directions — the resultant is not zero, and the forces are unbalanced. An unbalanced force changes the state of motion: it produces acceleration in the direction of the net force. If one team in the tug-of-war pulls harder, the rope and the weaker team move in the direction of the stronger pull. If the block on the table is pulled by a stronger string on the right, it moves to the right. When a footballer kicks a resting ball, the unbalanced force of the kick starts it moving; the ball then slows and stops because the unbalanced force of friction from the ground acts against its motion.

Friction deserves attention here because it explains why the old idea of force seemed true. Push a box across the floor: it moves while you push and stops when you stop. It seems that force is needed to keep it moving. In fact two forces act on the box — your push forward and friction backward. While your push exceeds friction, the net force is forward and the box accelerates; when your push exactly equals friction the forces are balanced and the box moves at constant speed; when you stop pushing, friction is the only force, unbalanced and backward, and the box decelerates to rest. Friction is the force that stops things, and if it could be removed a moving body would go on moving for ever. Galileo's thought experiment in the next topic does exactly that.

A summary the examination asks for: balanced forces — resultant zero, no change in state of motion, may change shape; unbalanced forces — resultant not zero, produce acceleration, change the state of rest or motion. Whether a body accelerates is decided not by how many forces act on it but by whether their sum is zero.

Galileo Galilei, in the early 1600s, studied the motion of balls on inclined planes and reasoned his way to a conclusion that overturned two thousand years of belief. A ball rolling down a slope speeds up; a ball rolling up a slope slows down. Now let a ball roll down one slope and up a second facing it: it climbs to nearly the same height it started from. Make the second slope less steep, and the ball travels farther along it to reach the same height. Make the second slope horizontal, and the ball, trying to reach a height it can never reach, would roll on for ever at constant velocity — if there were no friction. Since friction can never be removed completely in practice, the ball does eventually stop, but the less the friction, the farther it goes — a marble on a carpet stops soon, on a smooth floor it goes farther, on ice farther still. Galileo concluded that a body keeps moving with constant velocity when no net force acts on it; force is needed to change motion, not to keep it going.

Isaac Newton took Galileo's idea and made it the first of his three laws of motion, published in 1687 in the Principia. Newton's first law of motion states: An object remains in a state of rest or of uniform motion in a straight line unless compelled to change that state by an applied unbalanced force. In other words, all bodies resist a change in their state of motion; if at rest they tend to remain at rest, if moving they tend to keep moving at the same speed in the same direction. This tendency is called inertia, and the first law is therefore also called the law of inertia.

The law has two halves. The first — a body at rest stays at rest — matches everyday experience: a book on a table does not move by itself. The second — a body in motion keeps moving uniformly — seems to contradict experience, because moving things on Earth always stop; but they stop because of friction and air resistance, which are forces, not because motion runs out. In outer space, where there is no friction, a spacecraft coasts for years at constant velocity with its engines off, exactly as the law says.

The first law also gives a definition of force: force is that which changes, or tends to change, the state of rest or of uniform motion of a body. And it tells us when the net force on a body is zero — whenever the body is at rest or moving at constant velocity in a straight line. A car cruising at a steady 80 km/h on a straight highway has zero net force on it, even though its engine is working: the engine's push is exactly balanced by friction and air resistance. The moment the driver presses the accelerator or the brake, the forces become unbalanced and the velocity changes.

Inertia is the natural tendency of a body to resist any change in its state of rest or of uniform motion. All bodies have it, but not equally. It is harder to push a loaded truck than an empty one, harder to stop a rolling boulder than a rolling football, harder to move a full bucket than an empty one. The quantity that measures inertia is mass: the greater the mass of a body, the greater its inertia and the greater the force needed to change its motion. Mass is thus a measure of inertia, and its SI unit is the kilogram. This is the deepest meaning of mass, deeper than the amount of matter.

Inertia shows itself in three kinds of situation, and the examination asks for examples of each.

Inertia of rest — a body at rest tends to stay at rest. When a bus starts suddenly, passengers fall backward: their feet, in contact with the floor, move forward with the bus, but the upper body, by inertia, tends to stay at rest. When a carpet is beaten with a stick, the dust falls out: the carpet moves but the dust particles tend to remain at rest and separate from it. When a branch is shaken, ripe fruits and dry leaves fall. When a card resting on a tumbler with a coin on it is flicked, the card flies away but the coin, by inertia, drops into the tumbler. A tablecloth can be pulled quickly from under crockery, which stays in place.

Inertia of motion — a body in motion tends to keep moving. When a moving bus stops suddenly, passengers fall forward: their feet stop with the bus but their upper bodies continue to move. A runner who reaches the finishing line cannot stop at once. A passenger jumping from a moving bus falls forward, and must run a few steps in the direction of the bus's motion to avoid falling. An athlete runs some distance before a long jump to build up motion that carries her farther through the air. Mud sticking to a rotating wheel flies off along the tangent. A knife struck against a stone loosens the wet dough on it.

Inertia of direction — a body tends to keep its direction of motion. When a bus takes a sharp turn, passengers are thrown outward: their bodies tend to keep moving in the original straight line while the bus turns. Water on a spinning umbrella flies off along the tangent. Sparks from a grinding wheel fly off tangentially.

Seat belts and headrests. In a car crash the car stops in a fraction of a second but the passengers, by inertia of motion, continue forward and strike the windscreen or steering wheel. Seat belts hold the body to the seat and, being slightly elastic, spread the stopping force over a longer time. Headrests protect the neck when a car is struck from behind: the seat pushes the body forward but the head, by inertia of rest, stays back and would snap backward without support. These are the reasons the law requires belts and helmets.

A cricket ball moving slowly can be caught with bare hands; the same ball bowled fast can break a finger. A truck parked at the roadside is harmless; the same truck rolling slowly down a slope can crush a car. What matters in the impact of a moving body is not its mass alone or its velocity alone but the product of the two. This product is momentum, defined by Newton as the quantity of motion.

The momentum p of a body is the product of its mass m and its velocity v: p = mv. It is a vector quantity, in the direction of the velocity. Its SI unit is kilogram metre per second (kg m/s); there is no special name. A body at rest has zero momentum however large its mass, and a body of small mass at very high speed — a bullet — can have as much momentum as a heavy body moving slowly.

Worked example 1. A cricket ball of mass 0.16 kg bowled at 40 m/s: p = 0.16 × 40 = 6.4 kg m/s. A truck of mass 4000 kg rolling at 0.5 m/s: p = 4000 × 0.5 = 2000 kg m/s. A bullet of mass 0.02 kg at 300 m/s: p = 6 kg m/s — nearly the same as the cricket ball.

Worked example 2. Which has more momentum, a 1000 kg car at 20 m/s or a 10,000 kg truck at 1 m/s? Car: 20,000 kg m/s; truck: 10,000 kg m/s. The car, despite being ten times lighter.

Why is momentum, rather than velocity, the right measure of motion? Because the effect of a force depends on both mass and velocity. It takes more effort to stop a heavy body than a light one at the same speed, and more effort to stop a fast body than a slow one of the same mass — and momentum captures both. Newton found that the correct statement of his second law is in terms of momentum: the force acting on a body is proportional to the rate at which its momentum changes. A small force acting for a long time can produce the same change of momentum as a large force acting briefly — pushing a stalled car slowly up to speed, versus the sudden blow of a collision. This is developed in the next topic.

Change of momentum. If a body's velocity changes from u to v, its momentum changes from mu to mv, and the change is m(v − u). A ball of 0.5 kg thrown at 10 m/s and caught (v = 0) has a momentum change of 0.5 × (0 − 10) = −5 kg m/s; the catcher's hands must provide the force that produces this change. If the ball bounces back at 10 m/s the change is 0.5 × (−10 − 10) = −10 kg m/s, twice as much, because direction counts — momentum is a vector.

The idea that momentum is what a force changes, and the further idea that the total momentum of bodies acting on each other stays constant, are the two pillars of the rest of the chapter.

The first law says what happens when no net force acts. The second law says what happens when one does, and how much. Newton's second law of motion states: The rate of change of momentum of a body is directly proportional to the applied unbalanced force, and the change takes place in the direction of the force.

Derivation of F = ma. Consider a body of mass m moving with initial velocity u. An unbalanced force F acts on it for a time t and changes its velocity uniformly to v. Initial momentum = mu; final momentum = mv; change in momentum = mv − mu = m(v − u). Rate of change of momentum = m(v − u) / t. By the second law, F ∝ m(v − u) / t. But (v − u) / t is the acceleration a, so F ∝ ma, that is F = k ma, where k is a constant of proportionality. The unit of force is chosen so that k = 1: one unit of force is that which produces an acceleration of 1 m/s² in a mass of 1 kg. With this choice, F = ma. The SI unit of force is the newton (N), and 1 N = 1 kg m/s²: the force that gives a mass of 1 kg an acceleration of 1 m/s². The second law thus gives a way to measure force: force = mass × acceleration.

Notice three things in the equation. Force and acceleration are in the same direction. For a given force, the acceleration is inversely proportional to the mass — the same push accelerates a bicycle far more than a car, which is the quantitative form of inertia. And for a given mass, the acceleration is proportional to the force. If F = 0, then a = 0 and the velocity is constant: the first law is the special case of the second.

The momentum form, F = (mv − mu) / t = change in momentum ÷ time, is the more general statement and explains a set of everyday facts. The same change in momentum requires a smaller force if it is spread over a longer time. A fielder catching a fast cricket ball draws his hands backward with the ball, increasing the time over which the ball's momentum falls to zero and thereby reducing the force on his hands; a ball stopped abruptly hurts. A high jumper lands on a thick cushion and a long jumper on loose sand, which increase the stopping time and lower the force. Cars have crumple zones and air bags and passengers wear seat belts that stretch, so that in a crash the momentum is lost over a longer time and the force on the body is survivable. A karate expert striking bricks does the opposite — delivers the momentum change in the shortest possible time to produce a large force. Glassware is packed in straw or foam for the same reason. Time and force trade against each other for a fixed change in momentum.

The examination asks for the statement of the law, the derivation with the definition of the newton, and one of these applications explained in terms of time and force; the next topic supplies the numerical practice.

Problems on F = ma come in a few standard shapes: find F from m and a; find a and then use an equation of motion; find F from a change in momentum over a time; and find the stopping distance or time. Work in SI units — kilograms, metres, seconds, newtons — and convert grams to kilograms and km/h to m/s first.

Problem 1. A constant force acts on an object of mass 5 kg for 2 s and changes its velocity from 3 m/s to 7 m/s. Find the force. Then, if the force were applied for 5 s, what would the final velocity be? a = (7 − 3)/2 = 2 m/s²; F = ma = 5 × 2 = 10 N. For 5 s: v = u + at = 3 + 2 × 5 = 13 m/s.

Problem 2. Which would require a greater force — accelerating a 2 kg mass at 5 m/s² or a 4 kg mass at 2 m/s²? F₁ = 2 × 5 = 10 N; F₂ = 4 × 2 = 8 N. The 2 kg mass at 5 m/s² requires the greater force.

Problem 3. A motorcar of mass 1200 kg moving along a straight road at 90 km/h is brought to rest in 4 s by brakes. Find the change in momentum, the retardation and the braking force. u = 25 m/s, v = 0, t = 4. Change in momentum = m(v − u) = 1200 × (0 − 25) = −30,000 kg m/s. a = (0 − 25)/4 = −6.25 m/s². F = ma = 1200 × (−6.25) = −7500 N; the minus sign shows the force opposes the motion.

Problem 4. A hammer of mass 500 g moving at 50 m/s strikes a nail and is stopped in 0.01 s. Find the force of the nail on the hammer. m = 0.5 kg. F = m(v − u)/t = 0.5 × (0 − 50)/0.01 = −2500 N; the nail exerts 2500 N on the hammer, and the hammer 2500 N on the nail.

Problem 5. A force of 5 N gives a mass m₁ an acceleration of 10 m/s² and a mass m₂ an acceleration of 20 m/s². What acceleration would it give if both masses were tied together? m₁ = 5/10 = 0.5 kg; m₂ = 5/20 = 0.25 kg; together 0.75 kg; a = 5/0.75 = 6.67 m/s².

Problem 6. An automobile of mass 1500 kg is moving at 20 m/s; find the force needed to stop it in 50 m. Using v² = u² + 2as: 0 = 400 + 100a, a = −4 m/s². F = 1500 × (−4) = −6000 N, a retarding force of 6000 N.

Problem 7. A stone of 1 kg is thrown at 20 m/s across the frozen surface of a lake and comes to rest after travelling 50 m. Find the force of friction. a = (0 − 400)/(2 × 50) = −4 m/s²; F = 1 × (−4) = −4 N, friction of 4 N opposing the motion.

Problem 8. An 8000 kg engine pulls a train of five wagons, each of 2000 kg, along a horizontal track. If the engine exerts a force of 40,000 N and the track offers a friction force of 5000 N, find the net accelerating force, the acceleration of the train, and the force of wagon 1 on wagon 2. Net force = 40,000 − 5000 = 35,000 N. Total mass = 8000 + 10,000 = 18,000 kg; a = 35,000/18,000 = 1.94 m/s². Wagon 1 pulls wagons 2 to 5 (mass 8000 kg): force = 8000 × 1.94 = 15,556 N.

Problem 9. A bullet of 10 g travelling horizontally at 150 m/s strikes a wooden block and comes to rest in 0.03 s. Find the distance of penetration and the force of the wood on the bullet. a = (0 − 150)/0.03 = −5000 m/s²; s = ut + ½at² = 150 × 0.03 − ½ × 5000 × 0.0009 = 4.5 − 2.25 = 2.25 m; F = 0.01 × (−5000) = −50 N.

In each case the chain is: find a (from the equations of motion or from the given velocities and time), then F = ma; or find the change in momentum and divide by time. State the sign and say what it means.

The first two laws describe what happens to one body. The third describes what happens between two. When you push against a wall, the wall pushes back against your hand — you feel it. When you press a spring balance against a table, its reading shows the table pressing back. Newton generalised this into his third law of motion: To every action there is an equal and opposite reaction — whenever one body exerts a force on a second body, the second body exerts a force of equal magnitude and opposite direction on the first.

Four points prevent the usual confusions. 1. Forces always occur in pairs; a single isolated force does not exist. 2. The action and reaction are equal in magnitude and opposite in direction. 3. They act on different bodies — the action on one, the reaction on the other — and therefore they never cancel each other; forces cancel only when they act on the same body. 4. They act simultaneously; neither comes first, and either may be called the action.

Why, then, does anything move, if every force is matched by an opposite one? Because the two forces act on two different bodies, and each body responds to the force on it according to its own mass. When a gun fires, the gun pushes the bullet forward and the bullet pushes the gun backward with an equal force; the bullet, being light, gets a large acceleration and the gun, being heavy, a small one — but it does move, and that is the recoil.

Examples. Walking: the foot pushes the ground backward (action); the ground pushes the foot forward (reaction), and this forward push moves us. On ice or wet mud the ground cannot push back effectively, and we slip. Swimming: the swimmer pushes water backward; the water pushes the swimmer forward. A bird flying pushes air downward and backward with its wings; the air pushes the bird upward and forward. Rowing: the oars push water backward; water pushes the boat forward. A rocket or jet expels hot gases backward at high speed; the gases push the rocket forward, which is why a rocket works in empty space where there is nothing to push against — it pushes against its own exhaust. Recoil of a gun, described above. A balloon released with its neck open shoots forward as the air rushes out backward. A boat and a sailor: when a sailor jumps from a boat to the shore, the boat moves backward, because the sailor pushes the boat backward as the boat pushes the sailor forward. A book on a table: the book presses the table with its weight; the table pushes the book up equally. A hammer and nail: the hammer strikes the nail and the nail exerts an equal force back on the hammer, which is why the hammer stops.

A demonstration with two spring balances hooked together and pulled shows the law directly: both read the same, whichever is pulled. Two students on roller skates who push each other move apart in opposite directions, the lighter one faster. The third law is the basis of the law of conservation of momentum, which follows.

Combine the second and third laws and a powerful result follows: in an interaction between two bodies on which no external force acts, the total momentum stays the same.

Law of conservation of momentum: In the absence of an external unbalanced force, the total momentum of a system of bodies remains constant — the total momentum before an interaction (a collision, an explosion, a push) equals the total momentum after it. Momentum may pass from one body to another, but the sum is unchanged.

Derivation. Consider two balls A and B of masses m_A and m_B moving in the same straight line with initial velocities u_A and u_B, with u_A greater than u_B so that A catches up and collides with B. During the collision, which lasts a short time t, A exerts a force F_AB on B and, by the third law, B exerts an equal and opposite force F_BA on A: F_AB = −F_BA. After the collision the velocities are v_A and v_B.

By the second law, the force on A is its rate of change of momentum: F_BA = m_A(v_A − u_A) / t. Likewise the force on B: F_AB = m_B(v_B − u_B) / t.

Since F_AB = −F_BA: m_B(v_B − u_B) / t = −m_A(v_A − u_A) / t. Cancelling t and rearranging: m_Bv_B − m_Bu_B = −m_Av_A + m_Au_A, that is

m_Au_A + m_Bu_B = m_Av_A + m_Bv_B.

The left side is the total momentum before the collision and the right side the total momentum after it; they are equal. The momentum lost by A is exactly gained by B. The derivation used only the second and third laws, so conservation of momentum is not a new assumption but a consequence of Newton's laws; it holds whenever the only forces are the internal ones between the bodies, whatever the nature of the collision — bouncing apart, sticking together, or exploding.

What conservation means in practice. If two bodies start at rest — a gun and its bullet, a boat and a sailor, two skaters — the total momentum is zero before, so it must be zero after: the two bodies move off with equal and opposite momenta, and the lighter one moves faster. If one body strikes another at rest and they stick together, the combined body moves with a velocity that keeps the total momentum the same, and since the mass is larger the velocity is smaller. Momentum is a vector, so directions must be given signs — usually one direction positive and the other negative — and a negative answer means motion in the negative direction.

The examination asks for the statement, the derivation with the diagram of the two balls before, during and after collision, and a numerical example; these follow in the last two topics.

Recoil of a gun. Before firing, the gun and the bullet are at rest, so the total momentum of the system is zero. When the trigger is pulled, the expanding gases push the bullet forward and, by the third law, the gun backward; no external force acts on the gun-bullet system during the brief firing, so the total momentum must still be zero after firing. If the bullet of mass m leaves with velocity v and the gun of mass M recoils with velocity V, then 0 = mv + MV, so V = −mv / M. The minus sign shows that the gun moves opposite to the bullet. Because M is much larger than m, V is much smaller than v — a rifle recoils at about one metre per second while its bullet leaves at hundreds of metres per second — but the recoil is real and a shooter braces the butt against the shoulder to absorb it.

Worked example. A bullet of mass 20 g is fired horizontally at 150 m/s from a pistol of mass 2 kg. Find the recoil velocity of the pistol. m = 0.02 kg, v = 150 m/s, M = 2 kg. 0 = 0.02 × 150 + 2 × V; V = −3/2 = −1.5 m/s; the pistol recoils at 1.5 m/s in the direction opposite to the bullet.

Worked example. A gun of mass 100 kg fires a shell of 1 kg at 200 m/s. Recoil velocity = −(1 × 200)/100 = −2 m/s.

Rocket propulsion. A rocket carries fuel and an oxidiser; in the combustion chamber they burn to produce hot gases at very high pressure, which rush out of the nozzle at the rear at speeds of two to four kilometres per second. The gases carry momentum backward; to conserve the total momentum, the rocket gains an equal momentum forward. In the language of the third law, the rocket pushes the gases backward and the gases push the rocket forward. The rocket needs no air to push against, which is why it works in the vacuum of space where a jet engine or a propeller would be useless. The thrust depends on the mass of gas ejected per second and its speed; and as fuel is burnt the rocket's mass falls, so the same thrust gives a growing acceleration — which is why rockets are built in stages that are discarded when empty. India's PSLV and GSLV launch vehicles, and the Chandrayaan and Mangalyaan missions launched from Sriharikota, all ride on conservation of momentum. A toy balloon let go with its neck open, a garden sprinkler that spins as water jets out, and a boat that moves backward when a sailor jumps forward are humbler versions of the same principle.

Jet of water on a wall, the kick of a fire hose, the sideways push felt when a heavy bag is thrown from a stationary boat — each is an application of the same law: when part of a system is pushed one way, the rest goes the other way so that the total momentum is unchanged.

A caution about what conservation does not say: it does not say that each body keeps its momentum, only that the total does; nor does it apply if an external force acts during the interaction — friction from the ground on a gun mounted firmly, for instance, supplies an external force and the gun does not recoil freely. In problems the phrase no external force, or a very short interaction, is the signal to apply the law.

Set up every problem the same way: choose a positive direction; write the total momentum before, in kilograms and metres per second; write the total momentum after with unknowns; equate them; solve; interpret the sign.

Problem 1. Two objects of masses 100 g and 200 g move along the same line in the same direction with velocities 2 m/s and 1 m/s respectively. They collide, and after the collision the first object moves at 1.67 m/s. Find the velocity of the second. m₁ = 0.1 kg, m₂ = 0.2 kg. Momentum before = 0.1 × 2 + 0.2 × 1 = 0.2 + 0.2 = 0.4 kg m/s. Momentum after = 0.1 × 1.67 + 0.2 × v₂ = 0.167 + 0.2 v₂. Equating: 0.2 v₂ = 0.4 − 0.167 = 0.233; v₂ = 1.165 m/s in the original direction.

Problem 2. An object of mass 1 kg travelling at 10 m/s collides with a stationary wooden block of mass 5 kg and sticks to it; the two move together. Find the total momentum before and after, and the velocity of the combined object. Before: 1 × 10 + 5 × 0 = 10 kg m/s. After: by conservation, also 10 kg m/s. Combined mass 6 kg: v = 10/6 = 1.67 m/s.

Problem 3. A hockey ball of mass 200 g travelling at 10 m/s is struck by a stick so that it returns along the same path at 5 m/s. Find the change in momentum. Take the original direction as positive: initial momentum = 0.2 × 10 = 2 kg m/s; final momentum = 0.2 × (−5) = −1 kg m/s. Change = −1 − 2 = −3 kg m/s, that is 3 kg m/s in the direction of the return.

Problem 4. A girl of mass 40 kg jumps with a horizontal velocity of 5 m/s onto a stationary cart of mass 3 kg and sits on it. Find the velocity of the cart with the girl. Before: 40 × 5 + 3 × 0 = 200 kg m/s. After: (40 + 3) × v = 43 v. v = 200/43 = 4.65 m/s.

Problem 5. Two hockey players of opposite teams, one of 60 kg moving at 5 m/s and the other of 55 kg moving at 6 m/s in the opposite direction, collide and become entangled. In which direction and with what speed do they move? Take the first player's direction as positive. Before: 60 × 5 + 55 × (−6) = 300 − 330 = −30 kg m/s. After: 115 × v = −30; v = −0.26 m/s: they move in the direction of the second player at 0.26 m/s.

Problem 6. A truck of mass 3000 kg moving at 10 m/s collides with a car of mass 1000 kg moving at 5 m/s in the same direction; after the collision they move together. Velocity after? (30,000 + 5000)/4000 = 8.75 m/s.

Problem 7. Two objects each of mass 1.5 kg move in the same straight line but in opposite directions at 2.5 m/s each. They collide and stick together. Velocity after? Before: 1.5 × 2.5 + 1.5 × (−2.5) = 0. After: 3 × v = 0, so v = 0: the combined body is at rest, having equal and opposite momenta cancelled.

Problem 8. A bullet of 10 g is fired at 300 m/s into a wooden block of 990 g resting on a smooth surface and stays inside it. Find the velocity of the block. Before: 0.01 × 300 = 3 kg m/s. After: 1.0 × v = 3; v = 3 m/s.

Common errors: forgetting to convert grams to kilograms; giving both velocities the same sign when the bodies move in opposite directions; and adding masses when the bodies do not stick together. Always draw the before and after picture with arrows, and check that the answer's sign makes physical sense.

The three laws are best fixed by seeing them together, each with its statement, its key idea and its examples, and then applied to a few compound situations that the examination likes.

The law of conservation of momentum, derived from the second and third laws, says the total momentum of an isolated system is constant — the tool for collisions, explosions and recoil.

Compound applications. Why do we use helmets and seat belts? In a crash the vehicle stops in a fraction of a second; by the first law the rider or passenger continues forward (inertia of motion); the belt or the helmet's padding stops the body over a longer time, and by the second law the force is smaller (F = change in momentum ÷ time). Why is it dangerous to jump from a moving bus? The feet stop on touching the ground but the body, by inertia, keeps the bus's velocity and the person falls forward; running a few steps in the direction of motion lets the body slow gradually. Why does a cricketer pull his hands back? Second law, longer time, smaller force. Why does a gun kick? Third law and conservation of momentum. Why can a rocket fly in space? It pushes on its own exhaust, not on air. Why do passengers lurch outward on a turn? Inertia of direction; the seat and friction supply the inward force that turns them with the bus. Why is friction needed to walk? The ground can push us forward (third law) only if it grips the foot. Why does a heavy truck need bigger brakes than a car at the same speed? Greater mass means greater momentum and, for the same stopping time, greater force.

A note on friction and the first law. Friction appears in nearly every real example as the unbalanced force that stops things, and it is often blamed for making Newton's first law seem false. Reduce it — with wheels, oil, ice, air cushions — and moving bodies go farther; imagine it gone and they go for ever. The laws describe an ideal; friction is one of the forces that the laws must include, not an exception to them.

Units and symbols to carry into the examination: force F in newtons (N); mass m in kilograms (kg); acceleration a in m/s²; momentum p in kg m/s; 1 N = 1 kg m/s²; weight of 1 kg ≈ 9.8 N. The next chapter, gravitation, applies the second law to the force that the Earth exerts on everything near it.