NOTES : 1.
6 : : LINEAR MOMENTUM :
LEARNING OBJECTIVES :
Learning Outcomes : Candidate should be able to …….
(a) Understand the concepts of momentum and impulse.
(1.6) Momentum (b) Recall and use the equation momentum = mass × velocity, p=mv.
(c) Recall and use the equation for impulse Ft = mv – mu.
(d) Apply the principle of the conservation of momentum to solve simple problems in one
dimension.
➢ LINEAR MOMENTUM : - Momentum is defined as the product of mass and velocity.
❖ All the moving body has momentum & KE.
❖ Linear momentum = m × v
❖ Momentum has both magnitude and direction, it is a vector quantity.
❖ Its unit is kg m/ s.
❖ Slow moving body has smaller momentum than a fast moving body of same mass.
❖ If two bodies of unequal masses and velocities have same momentum, then their velocities are
inversely proportional to their masses.
𝐏𝟏 = 𝐏𝟐
𝐦𝟏 𝒗𝟏 = 𝐦𝟐 𝒗𝟐
𝒎𝟏 𝒗𝟐
=
𝒎𝟐 𝒗𝟐
➢ NEWTON’S SECOND LAW OF MOTION :
❖ Newton’s second law deals with the behaviour of objects on which all existing forces are not
balanced.
❖ STATEMENT : - It states that, “the rate of change of momentum of a body is directly proportional
to the external forces applied on it and the change in momentum takes place in the direction of the
force.
∆𝐏
𝐅∝
∆𝐭
∆𝐏
𝐅=𝒌
∆𝐭
Where, P = Change in momentum, t = Change in time and k = constant of proportionality.
Unit of force is chosen in such a manner that the constant k is equal to unity, i.e. k = 1.
∆𝐏 𝑭𝒊𝒏𝒂𝒍 𝒎𝒐𝒎𝒆𝒏𝒕𝒖𝒎 − 𝑰𝒏𝒊𝒕𝒊𝒂𝒍 𝑴𝒐𝒎𝒆𝒏𝒕𝒖𝒎 𝒎𝒗 − 𝒎𝒖
𝐅= = =
∆𝐭 𝒕 𝒕
𝒎 (𝒗 − 𝒖) 𝒗 − 𝒖
𝐅 = , 𝐁𝐮𝐭 = 𝒂 = 𝒂𝒄𝒄𝒆𝒍𝒓𝒂𝒕𝒊𝒐𝒏
𝒕 𝒕
F = ma
❖ The force acting on a body is measured by the product of mass of the body and acceleration produced
by the force acting on it. Thus, the second law gives us a measure of the force.
∆𝐏
❖ F = ma is a special case of 𝐅 = which only applies when the mass of the object is constant.
∆𝐭
❖ There are situations where the mass of the object changes as it moves, for example a rocket mass
changes continuously as it moves, because of its fuel burns.
➢ LAW/PRINCIPAL OF CONSERVATION OF MOMENTUM :
❖ STATEMENT : - It states that, “for a closed system, the total momentum in any direction is
constant, provided that there is no net external force acting on the system”.
❖ OR for a closed system, in any direction : total momentum of objects before collision = total
momentum of objects after collision.
❖ The total momentum of the system is zero.
➢ COLLISION :
➢ STATEMENT : - Collision means two objects coming into contact with each other for a very short
period. In other words, collision is a receprocative [exchange] interaction of energy/velocity between two
masses for a very short interval wherein the momentum and energy of the colliding masses changes along
with velocity. While playing carroms, you might have noticed the effect of a striker on coins when they
both collide.
➢ TYPES OF COLLISION :
❖ Three types of collision are : (i) Perfectly Elastic collision & (ii) Inelastic Collision & (iii) Perfectly
inelastic collision.
(i) PERFECTLY ELASTIC COLLISION: A collision in which both momentum and K.E. is
conserved is called perfectly elastic collision.
❖ In a perfectly elastic collision, no energy is converted into heat or other form of energy.
Consider a body A of mass m1 travelling with a speed u1 and another body B of mass m2
with a velocity u2. They collide elastically.
𝟏 𝟏
Their K.E. before collision are : of body A = 𝒎𝟏 𝒖𝟐𝟏 & of body B = 𝒎𝟐 𝒖𝟐𝟐
𝟐 𝟐
𝟏 𝟏
Their K.E. after collision are : of body A = 𝒎𝟏 𝒗𝟐𝟏 & of body B = 𝒎𝟐 𝒗𝟐𝟐
𝟐 𝟐
In a perfectly elastic collision since K.E. is conserved : KE before = KE after collision
𝟏 𝟏 𝟏 𝟏
𝒎𝟏 𝒖𝟐𝟏 + 𝒎𝟐 𝒖𝟐𝟐 = 𝒎𝟏 𝒗𝟐𝟏 + 𝒎𝟐 𝒗𝟐𝟐 ….. (1)
𝟐 𝟐 𝟐 𝟐
❖ In a perfectly elastic collision: Total energy, Momentum and K.E. of the system are
conserved.
Their momentum before collision are : of body A = 𝒎𝟏 𝒖𝟏 & of body B = 𝒎𝟐 𝒖𝟐
Their momentum after collision are : of body A = 𝒎𝟏 𝒗𝟏 & of body B = 𝒎𝟏 𝒗𝟐
In a perfectly elastic collision since momentum is conserved : momentum before =
momentum after collision.
𝒎𝟏 𝒖𝟏 + 𝒎𝟐 𝒖𝟐 = 𝒎𝟏 𝒗𝟏 + 𝒎𝟏 𝒗𝟐
❖ For all elastic collision, relative velocity of approach = relative speed of separation.
𝒖𝟏 − 𝒖𝟐 = 𝒗𝟐 − 𝒗𝟏
or, ½ m1u12 + ½ m2u22 = ½ m1v12 + ½ m2v22
❖ To find relative speed : we add velocities of two objects if they are travelling in opposite
direction and we subtract their velocities if they are travelling in the same direction.
(ii) INELASTIC COLLISION: - A collision in which momentum is conserved but KE is not
conserved is called inelastic collision.
❖ In an inelastic collision, some of K.E. is converted into heat energy.
(iii) PERFECTLY INELASTIC COLLISION: - A collision in which only moment is
conserved, and particles stick together after collision (i.e. move with the same velocity) is
called a perfectly inelastic collision.
❖ In an inelastic collision, K.E. is converted into heat energy.
➢ IMPULSE OF A FORCE [I] :
❖ Impulse of a force is defined as the product of the force and the time t for which it acts.
❖ It indicates, for how long the force was acting on the body.
❖ I = F × t = CHANGE IN MOMENTUM
❖ Its unit is Ns. (OR kg m/s)
❖ It is a vector quantity.
❖ For a variable force, the impulse I = Area under the Force – Time graph
❖ Impulse is equal in magnitude to the change in momentum of the body acted on by the force.
❖ Hence the change in momentum of the body is equal in magnitude to the area under a (net)
Force-Time graph. [incorrect to define impulse as change in momentum]
