Newton's Laws
Newton's First Law: A body will continue to remain in its state of
being - at constant velocity in a straight line or at rest - unless
acted upon by an external resultant force.
Newton's Second Law: The resultant force a body experiences is
directly proportional to the rate of change of momentum of the
body, and acts in the direction of the momentum change. This law
is commonly expressed in mathematics as
where
is
the resultant force acting on the body in the direction of the
acceleration,
is the mass of the body and
is the acceleration
of the body.
Newton's Third Law: If body A exerts a force on body B, body B
exerts a force of equal magnitude and opposite direction on body A.
Linear Momentum
Momentum is a vector quantity given by the product of its mass and
its velocity.
p is the momentum in kg ms-1 or Ns
m is the mass in kg
v the velocity in ms-1
Deriving F=ma
By Newton's Second Law Force is equal to the rate of change of
momentum.
Momentum is the product of mass and velocity.
If mass is constant this can be rewritten as:
is the rate of change of velocity or acceleration. Therefore when
mass is constant:
Principle of Conservation of Momentum
The Principle of conservation of momentum states that when
two particles collide:
total momentum before impact = total momentum after impact
m1u1 + m2u2 = m1v1 + m2v2
Elastic collisions
In elastic collisions both kinetic energy and linear momentum are
conserved. They do not exist in the real world but are idealised
scenarios which physicists use to simplify the mathematical of
models. For example, in kinetic molecular theory, it is assumed that
the collisions between gas molecules are fully elastic. (Eliminating
this assumption would make the model fiendishly difficult to
manipulate.)
In an elastic collision, the final and initial velocities of the colliding
particles must satisfy two conditions:
Solving for
and
we get:
Some rather elegant results emerge from these equations. For
example, if the masses of both colliding particles are equal, the
particles 'exchange' velocities upon impact. Furthermore, elastic
collisions have the property that
That is to say, the relative velocity of one particle with respect to
the other is reversed by the collision and the average of the
momenta before and after the collision is the same for both
particles. Mathematicians may view this simply as a generalization
of Newton's law of restitution (in the case where
Inelastic collisions
In an inelastic collision only linear momentum is conserved. Kinetic
energy is not conserved because as the bodies collide they suffer
energy losses in the form of heat dissipation. Nevertheless total
energy is always conserved. Physically speaking, inelastic collisions
are the only type of collision that feature in reality.
Collisions in one dimension in which the particles coalesce (merge
after collision) are inelastic. In this case, we can modify the
conservation of momentum to
Superelastic Collisions (Explosions)
A body of mass
is stationary and then 'explodes' to
produce two bodies, masses
and
moving in opposite
directions with velocities
and
. Alternatively, one is moving
with velocity
and the other is moving with velocity
.
Mathematically:
Kinetic Energy is clearly not conserved in this type of collision
either.
Impulse
Impulse is the change in momentum of a body and is equal to the
force applied to the body and the time for which it acts.
On a force-time graph, the Impulse is the area under the graph.