Impulse-Momentum Equation for Particles

Linear, not Angular, Momentum: In this section, we deal with linear momentum (mv) of particles only. Another section of

your book talks about linear (mvG) and angular (IGω) momentum of rigid bodies.

An Integrated Form of F=ma: The impulse-momentum (I-M) equation is a reformulation—an integrated form, like the work- energy equation—of the equation of motion, F=ma.

Particle Impulse-Momentum Equation Important! This is a Vector Equation!

mv12 + Fdt = mv

Initial Final Linear Momentum Linear Momentum = mv1 = mv “Impulse” 2 = Fdt Derivation of the Impulse-Momentum Equation

Derivation of the Impulse-Momentum Equation

Typical Eqn of Motion: F = ma dv Sub in a = dv : F = m dt dt If mass is changing: Net Force = d (e.g. for a rocket...) F = (mv) time rate of dt (Newton wrote it this way...) change of momentum

Separate variables: F dt = m dv t 2 v2 Set up integrals: F dt = m dv t1 v1

t2 Integrate: F dt = mv21 - mv t1

Usual form: mv12 + Fdt = mv “Impulse” vs. Particle Impulse-Momentum Equation

“Impulsive Force” mv12 + Fdt = mv

Initial “Impulse” Final Momentum Momentum = Fdt Impulse = Area under Force-time curve = Any force acting over Impulse = Fdt any time.... e.g. 100 lb for two weeks.... e.g. 1000 lb acting F 1000 lb Impulsive Force: in a 0.01 sec impact Area under A relatively large force F-t curve which acts over a very short Impulsive = 10 lb-sec t Force = Impulse period of time, like in an impact, 0.01 sec e.g. bat-on-ball, soccer kick, hammer-on-nail, etc. or 10 N acting Area under for 1 second... F 10 N F-t curve Non-Impulsive t = 10 N-sec Force 1 sec = Impulse Applications of the Impulse-Momentum Equation

For any problems involving F, v, t: The impulse momentum equation may be used for any problems involving the variables force F, velocity v, and time t. The IM equation is not directly helpful for determining acceleration, a, or displacement, s.

Helpful for impulsive forces: The IM equation is most helpful for problems involving impulsive forces. Impulsive forces are relatively large forces that act over relatively short periods of time, for example during impact. If one knows the velocities, and hence momenta, of a particle before and after the action of an impulse, then one can easily determine the impulse. If the time of impulse is known, then one can calculate the average force Favg that acts during the impulse. For problems involving graph of F vs. t: Some problems give a graph of Force vs. time. The area under this curve is impulse.

Important: You may need to find tstart for motion! Various Forms of the Impulse-Momentum Equation

Various Forms of the General Form: I-M Equation mv12 + Fdt = mv

Force During Impact: If force F is constant: Often mv12 + F t = mv Actual Force Modeled as AVG

F Know vectors v1 and v2:

FAVG F dt = mv21 - mv 0 t t = m( v21 - v ) Impulse Impulse = Area under curve... produces If force F is constant: change in ==F dt FAV G t momentum FAV G t = mv21 - mv

= m( v21 - v ) I-M Equation Compared to the Work-Energy Equation

Comparison Impulse-Momentum Eqn Work-Energy Eqn

dv Work: dU = F ds = m ads First write: F = m Both derived dt Eqn of Motion: F = ma t Kinematics: ad ds = v v from F=ma 2 v2 2 v2 Integrate: F dt = m dv Integrate: dU = mv dv v t1 1 1 v1

1 2 1 2 mv + U1-2 = mv Equation mv12 + Fdt = mv 2 1 2 2

1 1 or mv2 + F ds = mv2 2 1 2 2

Equation Impulse F dt produces Work U1-2 or Fds effects Explained a change in momentum, mv a change in KE, 1 2 2 mv

Vector or Scalar? A vector equation! A scalar equation!

Applications Problems involving v, F, t. Problems involving v, F, s. Not for problems needing Not for problems needing accel, a, or displacement, s. accel, a, or time, t.