How Forces Affect Motion

This chapter builds one connected story. It starts with force itself, how we recognise and measure it with a spring balance, then asks what happens when several forces act at once, introducing balanced and unbalanced forces and net force. Next comes friction, the force we usually forget because it has no visible push, yet it decides whether a pushed box moves and why a rolling ball eventually stops. Then come Newton's three laws in order: the first law (inertia, zero net force), the second law ($F = ma$), and the third law (equal, opposite forces on two different objects). It closes with a system of connected objects, such as two boxes joined by a string, where treating the whole system as one object of combined mass simplifies the problem.

What a force does, and how we measure it

A force can set an object moving from rest, speed it up, slow it down, change its direction, or change its shape. A kick sets a football moving, a cricket bat changes the direction of a ball's flight, and fingers squeezing a lemon change its shape. A force must always be described with both magnitude and direction, so force is a vector quantity, like position, displacement, velocity and acceleration. The SI unit of force is the newton, written with a small letter n, symbol N. We measure force with a spring balance: pulling its free end gives a reading equal to the pulling force, exactly as it reads the weight of an object when hung vertically.

Balanced and unbalanced forces, and net force

In real situations more than one force usually acts on an object at once. Push a box across a floor and friction also acts on it, opposite to the push. A ball floating on water feels gravity pulling it down and the buoyant force of the water pushing it up. To predict the object's motion we need the combined effect of all the forces, called the net force. Two forces equal in magnitude and opposite in direction are called balanced forces; the net force is zero and the object's motion does not change. If one force is larger, the forces are unbalanced: a net force acts in the direction of the larger force, equal to the difference of the two magnitudes. Two forces acting in the same direction simply add, giving a net force equal to their sum in that direction.

Friction, the force we usually forget

Push a box resting on the floor with a small force and it may not move at all, because the force of friction, arising between the box and the floor, acts opposite to the push and cancels it. Only when the applied force exceeds friction does a net force act, and the box starts to move. Once moving, if you stop pushing, friction alone continues to act opposite to the motion and gradually brings the box to rest — this is why a rolling ball or a coasting bicycle eventually stops. Friction depends on the nature of the two surfaces in contact: a stack of coins given a fixed push slides much farther on smooth polished tiles than on a rough wooden tabletop, because friction is smaller on the smoother surface. Pulling a block with a spring balance at a constant, just-moving speed gives a direct reading of the friction force, since applied force and friction are balanced at that instant. An object resting on a surface also has its weight (gravity, downward) balanced by the normal force (the surface's push, upward, perpendicular to it); these do not affect horizontal motion but must balance for the object to stay on the surface.

Newton's first law of motion (inertia)

For centuries it was mistakenly believed that a force is needed simply to keep something moving at constant velocity. Galileo Galilei argued, through thought experiments, that a body moving on a horizontal surface with no obstruction to its motion would carry on moving forever. Newton built on this and named the property inertia, the tendency of an object to resist any change in its state of rest or of uniform motion. Newton's first law states: an object at rest remains at rest, and an object in motion continues to move with a constant velocity, unless a net force acts upon it. Equivalently, zero net force means zero acceleration: a resting object cannot begin to move, and a moving object's velocity (magnitude and direction) cannot change. If friction were truly zero, a net force would still be needed to set a resting object moving, but once moving, no further force would be needed to keep it going at the same velocity forever.

Newton's second law of motion

The first law tells us what happens when net force is zero; the second tells us what happens when it is not. A gentle push on a ball gives a small acceleration, a strong push a large one, for the same ball. The same push also moves a light object faster than a heavy one: for the same force, a smaller mass gets a larger acceleration. Combining these two observations: acceleration is directly proportional to net force, and inversely proportional to mass.

where: $F$ is the net force in newtons (N), $m$ is the mass in kilograms (kg), and $a$ is the acceleration produced, in the same direction as $F$, in metre per second squared. One newton is the force that gives a 1 kg mass an acceleration of 1 m/s squared, so $1\ \text{N} = 1\ \text{kg} \cdot \text{m/s}^2$. A common special case is an object falling freely under gravity. The gravitational force on mass $m$, its weight, is

where: $g$ is the acceleration due to the Earth's gravity, about $9.8\ \text{m/s}^2$ (often rounded to $10\ \text{m/s}^2$), and $g$ does not depend on the object's mass; only the force $mg$ does. Newton's second law also explains why increasing the time over which a velocity change happens reduces the force needed. A fielder catching a fast ball pulls their hands back, stretching the time over which the ball's velocity drops to zero, lowering the acceleration and hence the force, reducing injury risk. Airbags and landing mats work the same way, by increasing stopping time to reduce the force.

Newton's third law of motion

At least two objects must interact for a force to arise. Kick a ball and you feel the ball pushing back on your foot. Newton's third law states this generally: whenever one object exerts a force on a second object, the second simultaneously exerts an equal and opposite force on the first, an action-reaction pair. Two identical spring balances hooked together and pulled always show the same reading, confirming the two forces are equal in magnitude no matter how hard you pull. The crucial subtlety is that the two forces of a third-law pair act on two different objects, so they never cancel, even though equal and opposite. When a canoeist pushes the paddle against the water, the water pushes the paddle (and canoe) forward with an equal force; one force acts on the water, the other on the paddle, and only the second affects the canoe's motion. The same idea explains walking (foot pushes ground backward, ground's friction pushes you forward), climbing a tree (legs push down the trunk, friction pushes you up), and rocket launches (engine expels gas downward, gas pushes the rocket upward). The third law applies to contact forces like friction and pushes, and to non-contact forces like gravity, magnetism and static electricity alike. Because a pair's forces act on objects of possibly very different mass, the resulting accelerations, $a = F/m$, are usually very different even though the forces are exactly equal — why the Earth and a falling fruit pull on each other equally, yet only the fruit's motion is noticeable, since the Earth's huge mass gives it a minuscule acceleration.

Forces on a system of connected objects

So far each law applied to a single object. Many situations involve objects connected together, such as a tractor pulling a trolley, or two boxes joined by a string. Consider boxes of mass $m_1$ and $m_2$ on a frictionless surface, connected by a string, with an external force $F$ pulling Box 1. The string exerts a force called tension on each box: Box 1 pulls Box 2 forward, and by Newton's third law, Box 2 pulls back on Box 1 with equal and opposite tension. Tension is an internal force of the system; it only shares the pull between the boxes. The only external force is $F$. Treating both boxes and the string as one combined object of mass $m_1 + m_2$, Newton's second law gives the whole system's acceleration directly, without ever finding the tension — often the fastest route to the answer.

Board level - Example 6.1: Combining two forces on a block

Two forces, 10 N and 6 N, act on a block on a table. Find the net force when the forces act (a) in the same direction, (b) opposite directions with 10 N larger, (c) opposite directions with the forces swapped so 6 N is now on the side 10 N was.

Step 1. Same direction: net force $= 10\ \text{N} + 6\ \text{N} = 16\ \text{N}$, in that common direction.

Step 2. Opposite directions, 10 N right and 6 N left: net force $= 10\ \text{N} - 6\ \text{N} = 4\ \text{N}$, towards the right.

Step 3. Swapped, 6 N right and 10 N left: magnitude unchanged, $4\ \text{N}$, but direction now follows the 10 N force, so it points left.

Answers: (a) 16 N, common direction. (b) 4 N, towards the 10 N side (right). (c) 4 N, towards the 10 N side (left).

Board level - Example 6.5: Pushing a block against friction

A student pushes a stationary block of mass 25 kg on a horizontal floor. Maximum friction opposing it is 50 N. Find the displacement in 2 seconds for a constant forward push of (i) 50 N, and (ii) 55 N.

Step 1. Case (i): applied force equals friction, 50 N each, so they balance and net force is zero. By Newton's first law the block stays at rest; displacement is zero.

Step 2. Case (ii): applied force exceeds friction: $F_{\text{net}} = 55\ \text{N} - 50\ \text{N} = 5\ \text{N}$ forward.

Step 3. By Newton's second law, $a = F_{\text{net}}/m = 5\ \text{N} / 25\ \text{kg} = 0.2\ \text{m/s}^2$.

Step 4. Using $s = ut + \tfrac{1}{2}at^2$ with $u = 0$, $a = 0.2\ \text{m/s}^2$, $t = 2\ \text{s}$: $s = \tfrac{1}{2}(0.2)(4) = 0.4\ \text{m}$.

Answers: (i) The block does not move; displacement = 0 m. (ii) The block moves 0.4 m forward.

Higher-order thinking (HOTS / olympiad-style) - A bullet stopping inside a wooden block

A bullet of mass 50 g moving at 100 m/s enters a heavy, stationary wooden block and comes to rest after penetrating a distance of 50 cm inside it. Assume the bullet decelerates uniformly while inside the block. (a) Estimate the average stopping force the block exerts on the bullet. (b) By Newton's third law, what force does the bullet exert on the block, and why does the massive block barely move even though this force is large?

Step 1. Convert to consistent SI units: mass $m = 50\ \text{g} = 0.05\ \text{kg}$, initial speed $u = 100\ \text{m/s}$, final speed $v = 0\ \text{m/s}$ (it stops), penetration distance $s = 50\ \text{cm} = 0.5\ \text{m}$.

Step 2. Since time is not given, use the kinematic equation connecting $u$, $v$, $s$ and $a$ without time: $v^2 = u^2 + 2as$. Substituting, $0 = (100)^2 + 2a(0.5)$, so $a = -10000\ \text{m/s}^2$; the magnitude of the deceleration is $10000\ \text{m/s}^2$.

Step 3. Apply Newton's second law for the average stopping force: $F = ma = 0.05\ \text{kg} \times 10000\ \text{m/s}^2 = 500\ \text{N}$, opposite to the bullet's motion.

Step 4. By Newton's third law, the bullet exerts an equal and opposite 500 N force on the block, forward along its original direction. These forces act on two different objects, so they never cancel.

Step 5. The block's acceleration is $a_{\text{block}} = F/m_{\text{block}}$. Since the block's mass is very much larger than the bullet's 0.05 kg, the same 500 N force gives it a correspondingly tiny acceleration — the same reasoning as the chapter's gun-and-bullet example (Example 6.8): equal third-law forces never imply equal accelerations when the masses differ.

Answers: (a) The average stopping force on the bullet is 500 N. (b) The bullet exerts an equal and opposite 500 N force on the block; the block's acceleration is tiny compared with the bullet's because its mass is much larger. Timing note: budget about 6 to 8 minutes for this two-part problem, roughly 3 minutes for the kinematics and force in part (a) and 3 to 5 minutes for the third-law and mass-acceleration reasoning in part (b); the block's exact mass is not given, only the qualitative comparison is required.

A recurring error is treating every force as if it acts alone: students forget friction, gravity and the normal force are usually present alongside any applied push. Always list every force on the object before deciding whether the net force is zero. Another common mistake is thinking a moving object needs a continuously applied force just to keep moving at constant velocity. Newton's first law says the opposite: with no net force, velocity stays unchanged forever; a continuous push is needed only to overcome friction. Students frequently mix up Newton's third law pairs with balanced forces. An action-reaction pair (a book pushing down on a table, the table pushing up on the book) acts on two different objects and can never cancel. Balanced forces act on the same single object (the book's weight and the normal force on the book) and can cancel to zero net force. Confusing these two is the single most common conceptual error in this chapter. A related slip is assuming equal and opposite third-law forces must produce equal accelerations. They never do unless the masses are equal, since $a = F/m$, as seen in the Earth-and-fruit and gun-and-bullet examples. Finally, students often forget to find the net force first before applying $F = ma$: using only one of several forces, or forgetting to subtract friction, gives a wrong acceleration even with correct arithmetic.

• Explain, with an everyday example, how a force can start, stop, speed up, slow down, or change the direction of motion, or change an object's shape.
• Distinguish balanced from unbalanced forces, and calculate the net force when two or more forces act on an object along the same line.
• Explain why friction can prevent a pushed object from moving, and why a moving object eventually comes to rest once the applied force is removed.
• State Newton's first law in your own words and use it to explain inertia, such as lurching in a vehicle that suddenly stops or starts.
• State Newton's second law as $F = ma$, and use it to find any one of force, mass or acceleration given the other two, including problems needing a kinematic equation first.
• Explain why increasing impact time (airbags, catching a ball, a landing mat) reduces the force in a collision.
• State Newton's third law, identify action-reaction pairs correctly, and explain why such pairs never balance, since they act on two different objects.
• Explain why equal and opposite third-law forces generally give very different accelerations, using $a = F/m$ and the objects' masses.
• Apply Newton's second law to connected objects (boxes joined by a string) by treating the system as one object of combined mass, and explain why internal tension cancels out.
• Read a velocity-time graph and determine, for each interval, whether the object has zero, forward, or backward net force, and calculate its magnitude given the mass.

Ask the tutor: If you are stuck on identifying which forces in a picture are an action-reaction pair versus which two forces are simply balanced on one object, ask the tutor to walk through a few labelled examples with you step by step.