(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 v_{0} then at any time t velocity of particle is given by the equation
v = v_{0} + 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 = mv_{0} + Ft
or
Ft = mv - mv_{0} -(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 t_{1} and t_{2},then impulse of the force is
I = F(t_{2}-t_{1}) -(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 t_{1} velocity of the particle is v_{1} and at time t_{2} velocity of the particle is v_{2},then
F(t_{2}-t_{1})=mv_{2}-mv_{1} -(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(t_{2}-t_{1}) -(5)
Linear momentum=p=mv -(6)
where
I=I_{x}i+I_{y}j+I_{z}k
F=F_{x}i+F_{y}j+F_{z}k
p=p_{x}i+p_{y}j+p_{z}k
v=v_{x}i+v_{y}j+v_{z}k
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(t_{2}-t_{1}) =mv_{2}-mv_{1} -(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 t_{1} velocity of the particle is v_{1} and at time t_{2} velocity of the particle is v_{2},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 (t_{2}-t_{1}) 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 t_{2}-t_{1} 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}
Example 1
A batsman hits back a ball straight in the direction of the bowler without changing its initial speed of 5 m/s
If the mass of the ball is 0.20 kg, determine the impulse imparted to the ball. (Assume linear motion of the ball)
Solution
Impulse =change in momentum
=.20 * 5 - (-.20* 5)
=2 N s
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