Momentum

1
CHAPTER 6

• Momentum
• and
• Collisions

2
Linear Momentum

• Newton formulated his version of dynamics based
  on the concept of momentum (p), which he defined
  as the product of the mass and velocity
• Momentum Mass ? Velocity
• p mv
• Because velocity is a vector quantity, so to is
  momentum.
• The direction of the momentum is in the direction
  of the velocity.
• Units kg?m/s

3
Linear Momentum

• Example
• Rich Gossage set a fastball record by hurling a
  0.14 kg baseball at a speed of 153 ft/sec. What
  was the magnitude of the balls momentum as it
  left his hand?

4
Linear Momentum

• Example
• A 0.149 baseball traveling at 28 m/s due south
  approaches a waiting batter. The ball is hit and
  momentarily crushed it springs back, sailing
  away at 46 m/s due north. Determine the
  magnitudes of it initial and final momenta and
  the change in its momentum.

5
Impulse and Momentum Change

• When a force is applied to a body, there will be
  a resulting proportional change in its motion.
• Newtons Second Law states that the net force
  applied to an object equals the resulting change
  in its momentum per unit time.
• Over a finite time interval, during which the
  force may change, Newtons Second Law is
• Fav ?p/?t

6
Impulse and Momentum Change

• Example
• A rocket fires its engine, which exerts an
  average force of 1000 N for 40 seconds in a fixed
  direction. What is the magnitude of the rockets
  momentum change?

7
Impulse and Momentum Change

• It is useful to combine the force and time into a
  single notion that equals the known momentum
  change.
• That single concept is called is called the
  impulse of the force, and an alternative
  statement of the Second Law becomes
• Fav?t ?p
• A given impulse will produce a specific change in
  momentum no matter what the mass or speed of the
  recipient body.

8
Impulse and Momentum Change

• If an object is originally at rest, it will take
  off in the direction of the net applied force,
  acquiring a momentum.
• Force applied to a body already in motion may
  either increase or decrease its momentum
  depending on whether the force acts parallel or
  antiparallel to the initial velocity.
• Example In order to turn a pitched baseball
  arriving at 90 mph into a homerun leaving at 110
  mph, a bat must apply a force of up to 8000 lbs.
  During this impact, which lasts only 1.25 ms,
  the ball is crushed to half of its diameter.

9
Impulse and Momentum Change

• Example
• On September 12, 1966, a Gemini spacecraft
  piloted by astronauts Pete Conrad and Richard
  Gordon met and docked with an orbiting Agena
  launch vehicle. With plenty of fuel left in the
  spacecraft, NASA decided to determine the mass of
  the Agena. While coupled, Geminis motor was
  fired, exerting a constant thrust of 890 N in a
  fixed direction for 7.0 s. As a result of that
  little nudge, the Gemini-Agena sped up by 0.93
  m/s. Assuming Geminis mass as a constant 34 ?
  102 kg, compute the mass of the Agena.

10
Varying Force

• Force may change from moment to moment.
• It might be easier to represent force by a curve
  (see diagram).
• The force starts out small as the object comes in
  contact with another object, rises to a maximum
  value when they are firmly in contact, and then
  drops off as the objects lose contact with each
  other.

11
Varying Force

• Just as the area under any portion of the speed
  time curve is the distance traveled during a
  given period, the area under the force-time curve
  is the impulse exerted during that time interval
  and it therefore equals the resulting change in
  momentum.
• Area ½?tFmax
• ?p ½?tFmax

12
Varying Force

• A real force-time curve can have a complex shape.
• Example A spaceship with a variable-thrust
  engine can generate a force-time curve with many
  bumps and wiggles, but the area encompassed in a
  given time interval will always be equal to the
  resulting change in momentum.
• When you throw a punch, your fist starts with pi
  0 and ends with pf 0. It accelerates up to a
  maximum speed and then decelerates to zero when
  your arm is extended.
• Given this fact, a karate blow is always aimed at
  a point inside the target so that it makes
  contact when p is maximum.

13
Varying Force

• Example
• A golfers club hits a 47.0 g golf ball from rest
  to a speed of 60.0 m/s in a collision lasting
  1.00 ms. The force on the ball rises to a peak
  value of Fmax and then drops to zero as it leaves
  the club. Compute a rough value for this maximum
  force by approximating the force-time curve, with
  a triangle of altitude Fmax.

14
Car Crashes

• When a car crashes into a brick wall, its front
  end deforms as it slows to a crushing stop.
• On average, automobiles compress roughly 1 inch
  for every mile per hour of speed just prior to
  impact.
• If we assume that the collision-deceleration is
  fairly uniform, the crush distance (sc) divided
  by vav ½(vi vf), with vf 0, is the impact
  time, thus
• ?t sc/vav
• ?t 2sc/vi

15
Car Crashes

• Since tests show that sc is proportional to vi,
  the impact time for cars of comparable stiffness
  should be independent of the speed of impact.
• A typical head-on, brick-wall collision lasts
  around 100 ms (airbags inflate in 55 ms).
• Crashing head-on into an identical car traveling
  toward you at the same speed is effectively the
  same as hitting a stationary brick wall.

16
Car Crashes

• Example
• A 70-kg passenger riding in a typical automobile
  is involved in a 40 mph head-on collision with a
  concrete barrier. Taking the stopping time as 100
  ms, compute the average force exerted by the
  seatbelt and shoulder strap on the person.

17
Jets and Rockets

• Imagine yourself on roller skates holding a bag
  of oranges.
• Now you throw one of them due north, and away you
  go due south. Why?
• You push on the orange in the forward direction
  during the throw, it pushes back with the same
  impulse on you, and back you go.
• A rocket works by hurling out a tremendous number
  of tiny high-speed (3 to 4-km/s) objects
  (molecules).
• Example During launch, the solid fuel boosters
  for the Space Shuttle expel 8.5 tons of fiery
  exhaust each second. The engines blast exhaust
  downward, and the escaping gas, in turn, pushes
  the up on the object (action/reaction).

18
Jets and Rockets

• Example
• A rocket engine testing a low-power fuel expels
  5.0 kg of exhaust gas per second. If these
  molecules are ejected at an average speed of 1.2
  km/s, what is the thrust of the engine?

19
Conservation of Linear Momentum

• Conservation of Momentum
• When the resultant of all the external forces
  acting on a system is zero, the linear momentum
  of the system remains constant (pi pf).
• The net momentum of an isolated system cannot
  change.
• The total momentum of a system of interacting
  masses must remain unaltered, provided that no
  net external force is applied.

20
Conservation of Linear Momentum

• Example
• According to published figures, a bullet fired
  from a standard 9-mm Luger pistol has a mass of
  8.0 grams and a muzzle speed of 352 m/s. If the
  mass of the gun is 0.90 kg, what is the recoil
  speed when fired horizontally?

21
Conservation of Linear Momentum

• Since neither the gun nor bullet were moving
  prior to firing the pistol, they have no initial
  momentum.
• pi pf
• pi 0 pf
• 0 mbvbf mgvgf
• 0 (0.0080 kg)(352 m/s) (0.90 kg)vgf
• vgf -(0.0080 kg)/(0.90 kg) ? 352 m/s
• vgf -3.1 m/s
• The (-) sign indicates that the gun is moving in
  the opposite direction of the bullet.

22
Conservation of Linear Momentum

• Example
• While floating in space a 100-kg robot throws a
  0.800 kg wrench at 12.0 m/s toward his partner
  working on the spaceship. How fast will the robot
  move away from the ship?
• pi pf
• 0 mrvrf mwvwf
• 0 (100 kg)vrf (0.800 kg)(12.0 m/s)
• vrf -0.096 m/s
• vrf -0.1 m/s

23
Collisions

• A collision is marked by the transfer of momentum
  between objects in relative motion resulting from
  their interaction via at least one of the four
  forces.
• In all cases where there are no external forces
  acting, the total momentum of the colliding
  objects is conserved.
• There are two types of collisions that we will be
  discussing in this chapter.

24
Collisions

• Inelastic Collisions
• An inelastic collision is one where the final
  kinetic energy (KE ½mv2) of the system is
  different from the initial KE.
• KEi ? KEf
• Example
• When you drop a tennis ball to the floor, it
  momentarily comes to rest, and then springs back,
  popping into the air. But the squashing of the
  ball produces some internal heating, and the ball
  only returns about two-thirds of the way back up.
• All collisions between macroscopic objects are
  more or less inelastic.
• The completely inelastic collision is at one
  extreme where the impacting objects stick
  together and the maximum amount of KE is
  transformed (i.e. lost) into internal energy.
• m1v1i m2v2i (m1 m2)vf

25
Collisions

• Example
• During a rainy day football game, a 854 N
  quarterback is standing holding the ball looking
  for a receiver when hes unkindly hit by a 1281 N
  tackle charging in at 6.1 m/s.
• At what speed do the two men, tangled together,
  initially sail off on the wet field? (Disregard
  friciton)
• How much mechanical energy is lost to friction?

26
Collisions

• pi pf
• mqvqi mtvti (mq mt)vf
• 0 (1281/9.81)(6.1) ((854/9.81)
  (1281/9.81))vf
• 796.8 217.7vf
• vf 3.7 m/s
• KEi ½mqvq2 ½ mtvt2
• KEi 0 ½(1281/9.81)(6.1)2
• KEi 2.4 kJ
• KEf ½(mq mt)vf2
• KEf ½((854/9.81) (1281/9.81))(3.7)2
• KEf 1.5 kJ
• So, 2.4 1.5 0.9 kJ of energy was lost.

27
Collisions

• For completely inelastic collisions between two
  bodies, only one of which (m1) is moving, the
  ratio of the total final KE to the total initial
  KE is
• KEf (m1/(m1 m2))KEi

28
Collisions

• 2. Elastic Collisions
• A collision is elastic when KE is constant.
• KEi KEf
• KE1i KE2i KE1f KE2f
• The relative speeds of the two bodies before
  and after an elastic impact are equal.
• The relative velocity is reversed by the
  collision, but its magnitude is unchanged.
• This is true for all elastic collisions.
• Example Newtons Cradle.

29
Collisions

• Example
• Two billiard balls move toward one another. The
  balls have identical masses, and assume that the
  collision between them is perfectly elastic. If
  the initial velocities of the balls are 30.0
  cm/s and -20.0 cm/s, what is the velocity of each
  ball after the collision.?

30
Collisions in Two Dimensions

• When two objects with masses m1 and m2
  elastically slam into one another off center
  (glancing collision) they subsequently move away
  at different angles to the original direction.
• Example Playing pool.
• Provided that there are no external forces
  acting, momentum is always conserved.
• Because of the 2-D of the collisions, we will
  make use of the fact that the scalar momentum
  components in the x and y directions are
  conserved independently.
• pix pfx and piy pfy

31
Collisions in Two Dimensions

• In a glancing collision, when the two objects
  have an equal mass (m1 m2), and one of them is
  at rest, the two will always move off at right
  angles to each other.
• We will only be dealing with objects in glancing
  collisions that stick together after the
  collision in order to avoid having too many
  unknowns.

32
Collisions in Two Dimensions

• Example
• Two cars enter an icy intersection and skid into
  each other. The 2.50 103 kg sedan was
  originally heading south at 20.0 m/s, whereas the
  1.45 103 kg coupe was driving east at 30.0 m/s.
  On impact, the two vehicles become entangled and
  move off as one at an angle (?) in a
  southeasterly direction. Determine the angle and
  the speed at which they initially skid away after
  crashing.