Newton’s laws of motion (developed by Isaac Newton in the 17^th century) describe how an object behaves under the action of applied forces.  The three laws can be stated as follows:

Law 1:             1.  A body continues in its state of uniform motion if the total force acting on the body is zero.

Law 2:             2.  The rate of change of momentum of a body is equal to the total force acting on that body.

Law 3:             3.  To every action there is an equal and opposite reaction.

There is nothing counter-intuitive about these laws.  As an example of the first law, imagine driving along at 30 mph in your car and then slamming on the brakes.  Until the friction force between your pants and the car seat starts to act, you are thrown forward in the car.  If the friction force is not sufficient then you are thrown against the steering wheel.  That is, your body wants to continue in its state of uniform motion – until the friction force starts to act on it.

The third law is one of the most quoted scientific principles.  One example of this is the explanation for why we don’t fall through floors.  The weight of our body applies a force to the floor, but the floor exerts an equal and opposite force on our body, thus balancing our weight and keeping us from falling downwards.  Another example is when one billiard ball hits another.  The first billiard ball applies a force to the second one, thus changing the state of motion of the second ball.  However, the second ball exerts an equal and opposite force on the first one, thereby changing the state of motion of the first ball.

For the current purpose, however, the most important of these is Newton’s second law, which, in the form quoted above, states that

            applied force = rate of change of momentum

The momentum of an object is equal to its mass M multiplied by its velocity v:

            P = Mv

The units of momentum (in SI units) are kg m/sec.  If the mass of the object remains constant (which it usually does, unless one is considering the dynamics of a space rocket), then the second law becomes:

            applied force = mass ´ acceleration

where ‘acceleration’ is defined as

            acceleration = rate of change of velocity

When we talk about ‘rate of change of velocity’, we are referring to the change in velocity in unit time.  Thus, if the speed of our car changes from 30 mph to 35 mph in 5 seconds, then the rate of change of velocity is (35-30)/5 = 2.5 mph per second.

Note that the applied force that appears on the left hand side of these expressions is the sum of all the forces acting on the object.  For a car, these would be:

            applied force = tractive force from engine - frictional force - aerodynamic drag

The minus signs appear in this expression because the frictional force and aerodynamic forces act in the opposite direction to the tractive force from the engine.  When the applied force is zero, the acceleration is zero.  When the car reaches its top speed, the applied force is zero and the frictional and aerodynamic forces exactly balance the tractive force from the engine.

If the applied force is constant at all times, then the calculation of velocity as a function of time (and hence the time required to accelerate from one velocity to another) is easy.  If the total applied force is F, then from (3) the acceleration is simply F divided by M, and the velocity as a function of time is

where t is the time for which the force is applied, and v₀ is the starting velocity.

However, if the applied force changes with time, calculation of velocities as a function of time becomes more difficult and the second law has to be rewritten slightly:

In this expression, dv is the (small) change in velocity that results over a very small time period dt (the ‘d’ that precedes the v and t is a standard notation that means that the changes in v and t are small).  For example, if dt is 0.001 seconds, dv is 0.003 m/sec and M is 1000 kg, then the applied force F is 1000 ´ 0.003/0.001 = 3000 newtons.  This form of Newton’s second law will be used later.  Note that in this notation,

Note also that the standard unit of force is the ‘newton’ – equal to the force required to accelerate a mass of 1 kg at an acceleration of 1 metre/sec/sec.  The newton is therefore equal to the weight of an object of mass 100g (approximately).

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