Newton's three laws

You let go of a trolley, yet it keeps moving. With a load added, the same push makes it gain speed more slowly. Does force determine whether something moves, or how its velocity changes?

Push a trolley across a rough floor, then imagine making the floor smoother and smoother. The smaller the resistance, the longer the motion continues. The limiting model is a free object: in an inertial frame, it remains at rest or moves in a straight line at constant velocity. This is Newton's first law. A trolley stops because of interactions such as friction, not because it uses up a stored supply of push.

Why must we specify the observer? In a braking bus, an unsecured ball starts moving through the cabin even though nobody pushes it forwards. A frame in which a free object obeys the law of inertia is called inertial. Another frame moving uniformly in a straight line relative to it, with no rotation of its axes, is also inertial. In ordinary school problems, a laboratory on Earth is treated as approximately inertial.

To find the cause of acceleration, first circle the object you are studying. What acts on it: Earth, the floor, a rope, another trolley? Each interaction contributes a force on that object. Leave the reaction forces acting on other objects outside this diagram for now. Otherwise, you may add forces with different recipients and wrongly predict equilibrium for a trolley that is actually accelerating.

Two ropes pull a trolley in opposite directions. If one pulls right with 7 N and the other left with 3 N, their combined effect is 4 N to the right. The net force is the vector sum of all forces on one object. Increasing the leftward pull to 7 N gives zero net horizontal force. Check the vertical forces separately: the absence of vertical motion is not a reason to ignore them without checking their sum.

Compare an empty trolley and a loaded one under the same net force. Over the same time, the loaded trolley's velocity changes less. This resistance to a change in velocity is inertia; its measure is mass m in kilograms. Doubling the mass at the same force halves the acceleration. Mass does not oppose motion itself: without a net force, a heavy moving object keeps its velocity just as a light one does.

Experiments with force and mass lead to one relation: ΣF⃗ = ma⃗, or Fₓ = maₓ along a chosen axis. This is Newton's second law for constant mass in an inertial frame. Double the net force and acceleration doubles; double the mass and acceleration halves. Here, 1 Н = 1 кг·м/с². The law relates force to a change in velocity, so the initial velocity is also needed to predict the motion itself.

Imagine the same force directed along, against and perpendicular to velocity. In the first case it increases speed, in the second it decreases speed, and in the third it changes direction. Acceleration always points along the net force; velocity need not. If the forces sum to zero, the second law gives a⃗ = 0, so velocity stays constant. Rest is the special case in which that constant velocity is zero from the start.

Two skaters push off each other. Can one push the other without feeling any response? Newton's third law says their interaction forces are of the same kind, equal in magnitude and opposite in direction. If their masses differ, equal forces produce different accelerations according to a = F/m. The lighter skater therefore changes velocity more, although the force acting on them is no greater than the reaction force they exert.

But equal and opposite forces should cancel, so why do both skaters move apart? Label the recipient of each arrow. One force acts on the first skater and the other on the second; both do not belong in one object's equation. If the pair is treated as a system, these internal forces cancel in the total. For one object to be in equilibrium, the forces acting on that particular object must balance.

A trolley of mass 2 kg is pulled right with a force of 9 N. A resistance force of 3 N acts left. The vertical forces balance. Find the acceleration.

Isolate the trolley and choose an axis pointing right. The forces on it are a 9 N pull to the right and 3 N resistance to the left. The problem states that the vertical forces sum to zero.

First find the combined effect: Fₓ = 9 − 3 = 6 Н. This sum determines the change in velocity.

aₓ = Fₓ/m = 6/2 = 3 м/с². The positive sign means the acceleration points right.

If resistance rose to 9 N, acceleration would be zero. If the mass doubled while the forces stayed the same, acceleration would halve. Both checks agree with the physical meaning of the law.

The trolley's acceleration is 3 m/s² to the right: its velocity component increases by 3 m/s each second while the forces remain unchanged.

With no net force, an object keeps its velocity. A continuing force is not needed just to keep it moving.

The sum of the forces determines acceleration; mass determines how much velocity changes under that force.

Interactions always involve both objects, but the paired forces act on different objects.

Next, apply Newton's laws to gravity, springs, scales and friction.