Linear momentum

(1) Introduction

  • We have already studied about the Newton's Laws of Motion and about their application
  • It becomes difficult to use Newton's law of motion as it is while studying complex problems like collision of two objects,motion of the molecules of the gas,rocket propulsion system etc.
  • Thus a further study of newton's law is required to find some theorem or principles which are direct consequences of Newton's law
  • We have already studied one such principle which is principle of conservation of energy.Here in this chapter we will define momentum and learn about the principle of conservation of momentum .
  • Thus we begin this chapter with the concept of impulse and momentum which like work and energy are developed from Newton's law of motion

(2) Linear momentum

  • From our daily life experiences like during the game of table tennis if the ball hits a player it does not hurt him. On the other hand, when a fast moving cricket ball hits a spectator, it may hurt him.
  • This suggests that impact produced by moving objects depends on both their mass and velocity.
  • So, there appears to exist some quantity of importance that combines the object's mass and its velocity called momentum and was introduced by Newton.
  • Momentum can be defined as "mass in motion". All objects have mass; so if an object is moving, then it has momentum - it has its mass in motion.
  • The momentum, p of an object is defined as the product of its mass, $m$ and velocity, $v$. That is, Momentum $p=mv$           (1)
  • Momentum has both direction and magnitude so it is a vector quantity. Its direction is the same as that of velocity, $v$.
  • The SI unit of momentum is kilogram-meter per second (kg m s-1).
  • Since the application of an unbalanced force brings a change in the velocity of the object, it is therefore clear that a force also produces a change of momentum.
  • We define the momentum at the start of the time interval is the initial momentum and at the end of the time interval is the final momentum.
  • When the object moves then it gains momentum as the velocity increases. Hence greater the velocity greater is the momentum.
  • Newton actually expressed his second law in momentum form
    "Force is directly proportional to the rate of change of momentum of a body and it takes place in the direction in which the force acts."
  • $F = \frac {dp}{dt}$
    $F =\frac {d(mv)}{dt}$
    for constant mass
    $F= m \frac {dv}{dt} =ma$
  • So $F = \frac {dp}{dt}$ and $F= ma$ are equivalent
  • This can be expressed in vector form as
    $f= \frac {d\mathbf{p}}{dt}$

Example -1
Find the momentum of the object of mass of .5 kg and moving with velocity 10 m/s?
Momentum is given by
$p= mv = .5 \times 10 = 5 kgms^{-1}$
Example -2
A tennis player hit the ball at the Speed 40 m/s . The mass of the ball is .05 kg and it was in contact with racket for 4 ms. Find the force applied by the tennis player?
$F = \frac {dp}{dt}$
Initial momentum =0
Final momentum= .05 x 40 = 2 kgm/s
$F = \frac {2 -0}{4 \times 10^{-3}} = 500 N$
Watch this tutorial for more information on How to solve momentum problems

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