Impulse and momentum

(3) Impulse and Linear momentum

• To explain the terms impulse and momentum consider a particle of mass m is moving along x-axis under the action of constant force F as shown below in the figure
  
  
  
• If at time t=0 ,velocity of the particle is v₀ then at any time t velocity of particle is given by the equation
  v = v₀ + at
  where a = F/m
  can be determined from the newton's second law of motion .Putting value of acceleration in above equation\
  we get
  mv = mv₀ + Ft
  or
  Ft = mv - mv₀                   -(1)
  
• right side of the equation Ft, is the product of force and the time during which the force acts and is known as the impulse
  Thus
  Impulse= Ft
  
• If a constant force acts on a body during a time from t₁ and t₂,then impulse of the force is
  I = F(t₂-t₁)                  -(2)
  Thus impulse received during an impact is defined as the product of the force and time interval during which it acts
  
• Again consider left hand side of the equation (1) which is the difference of the product of mass and velocity of the particle at two different times t=0 and t=t
  
• This product of mass and velocity is known as linear momentum and is represented by the symbol p. Mathematically
  p = mv                  --(3)
  
• physically equation (1) states that the impulse of force from time t=0 to t=t is equal to the change in linear momentum during

• If at time t₁ velocity of the particle is v₁ and at time t₂ velocity of the particle is v₂,then
  F(t₂-t₁)=mv₂-mv₁                  -(4)
  
• so far we have considered the case of the particle moving in a straight line i.e along x-axis and quantities involved F,v, and a were all scalars
  
• If we call these quantities as components of the vectors F,v and a along x-axis and generalize the definitions of momentum and impulse so that the motion now is not constrained along one -direction ,Thus we got
  Impulse=I=F(t₂-t₁)                  -(5)
  Linear momentum=p=mv                  -(6)
  where
  I=I_xi+I_yj+I_zk
  F=F_xi+F_yj+F_zk
  p=p_xi+p_yj+p_zk
  v=v_xi+v_yj+v_zk
  are expressed in terms of their components along x,y and z axis and also in terms of unit vectors
• On generalizing equation (4) using respective vectors quantities we get the equation
  F(t₂-t₁) =mv₂-mv₁                  -(7)
• So far while discussing Impulse and momentum we have considered force acting on particle is constant in direction and magnitude
  
• In general ,the magnitude of the force may vary with time or both the direction and magnitude may vary with time
  
• Consider a particle of mass m moving in a three-dimensional space and is acted upon by the varying resultant force F. Now from newtons second law of motion we know that
  F=m(dv/dt)
  or Fdt=mdv
  
• If at time t₁ velocity of the particle is v₁ and at time t₂ velocity of the particle is v₂,then from above equation we have
  
  
  
• Integral on the left hand side of the equation (8) is the impulse of the force F in the time interval (t₂-t₁) and is a vector quantity,Thus
  
  Above integral can be calculated easily if the Force F is some known function of time t i.e.,
  F=F(t)
  
  
• Integral on the right side is when evaluated gives the product of the mass of the particle and change in the velocity of the particle
  
  
• using equation (9) and (10) to rewrite the equation (8) we get
  
  
• Equivalent equations of equation (11) for particle moving in space are
  
  
• Thus we conclude that impulse of force F during the time interval t₂-t₁ is equal to the change in the linear momentum of the body on which its acts

• SI units of impulse is Ns or Kgms^-1