IMPULSE AND MOMENTUM When ever things collide, I’ve heard, Momentum is the chosen word. And if the crash is quick, You’ll see – It’s Impulse! (that’s dp/dt)

LINEAR MOMENTUM For an individual mass we define the linear momentum to be: From the 2nd law we have:

Momentum MOMENTUM IS: SYMBOLIZED BY p MEASURED IN UNITS OF kg m/s MASS x VELOCITY CONSERVED (FOR A CLOSED, ISOLATED SYSTEM) A VECTOR THE DIRECTION OF AN OBJECT’S MOMENTUM IS THE SAME AS THE VELOCITY

Momentum There are two general types of momentum problems: Explosions Guns, objects connected by springs, nuclear decay, etc Total momentum is always zero (both objects initially at rest) Collisions Elastic (both p and K are conserved) Inelastic (p conserved, TOTAL ENERGY is conserved) Totally inelastic (p conserved, TOTAL ENERGY is conserved)

IMPULSE For a constant force, the impulse is defined as the force multiplied by the time over which it acts If the force is not constant, the impulse is defined by an integral:

IMPULSE Impulse is defined as the change in momentum: J = Dp According to Newton’s second law (F = ma), a force is required to change velocity (a = dv/dt). Since this force acts during some finite time interval (F = m dv/dt). Impulse is equal to the product of the force and the time period, Fdt = m dv Therefore: Fdt = dp The units of impulse are the same as the units of momentum and may be expressed as: kgm/s or Ns

IMPULSE The area under the curves in the diagram represents the impulse on an object. Impulse may also be calculated by determining the change in momentum before and after a collision

EXAMPLE PROBLEM A 5kg mass moving at 10m/s in the +x direction is acted upon by a force in the –x direction with a magnitude given as a function of time by the graph. Determine the velocity of the mass after the force has stopped acting Force (x 103N) 8 6 4 2 0 2 4 6 8 10 12 14 Time (x 10-3 s)

solution The initial momentum of the object is: 50 kgm/s The area under the curve is: - 80 N s. The final momentum is: -30 kgm/s The final velocity is: -6m/s

CONSERVATION OF LINEAR MOMENTUM If no net external force acts on a system of particles, the total linear momentum of the system cannot change. (CH. 9 #40) Valid for objects moving or initially at rest Momentum is a vector (CH 9 #22) In two, or three, dimensions, if the net external force on a closed system is zero along an axis, then the component of the linear momentum along that axis cannot change. (CH. 9 #20)

Elastic Collisions in One Dimension BOTH OBJECTS MOVING Momentum is conserved

Elastic Collisions in One Dimension So is kinetic energy! Ko = Kf

Elastic Collisions in One Dimension THE RESULT OF THIS? WHICH LEADS TO: WHICH BECOMES: The relative velocity changes direction but keeps same magnitude.

EXAMLE ONE A baseball with a speed of -40m/s is hit by a bat with a speed of 18m/s. If the ball leaves the bat with a speed of 75m/s, what is the speed of the bat just after the collision? (assume the collision is completely elastic)

YOU MUST BE ABLE TO DERIVE EQUATIONS (9-67), ALGEBRA ALERT! YOU MUST BE ABLE TO DERIVE EQUATIONS (9-67), (9-68), (9-75), AND (9-76)

Inelastic Collisions When objects collide and stick together – momentum and total energy is conserved but kinetic energy is not. Some of the kinetic energy is transferred into _______ EXAMPLE: The ballistic pendulum

THANK YOU FOR YOUR INTEREST IN PHYSICS!