Category: 03 Dynamics

3-3-2-2 Elastic Collisions

If ONE BALL rams into the stack, why must ONE BALL come out the other end at the same speed? Why can’t TWO BALLS come out at HALF the speed, for example? (This outcome also conserves the total momentum!) And how about all FIVE BALLS swinging away at 1/5 the speed?

True enough. There are infinite number of outcomes that fulfil the Principle of Conservation of Momentum! However, since these collisions are elastic, the total kinetic energy should be unchanged after the collision as well. With this additional constraint, there is only one possible outcome. The same number of balls always come out the other end at the same end, conserving both total momentum and keeping total kinetic energy unchanged.

Let’s now bring Newton and PSY together for an xmphysics-style pendulum.

3-3-2-4 “Classic” Collisions

(Perfectly) Elastic “Sitting Duck” Collisions

canonicalA.gif

  • For all the above,
    • the total momentum is conserved.
    • when one mass is much larger than the other, the velocity of the heavier mass is hardly changed by the collision.
    • The two masses approach and separate from each other at relative speed of u.

 

Perfectly Inelastic “Sitting Duck” Collisions

canonicalB

  • For all the above,
    • the total momentum is conserved.
    • when one mass is much larger than the other, the velocity of the heavier mass is hardly changed by the collision.
    • The two masses travel at a common velocity after the collision.

(Perfectly) Elastic “Jousting” Collisions

canonicalC

  • For all the above,
    • the total momentum is conserved.
    • when one mass is much larger than the other, the velocity of the heavier mass is hardly changed by the collision.
    • The two masses approach and separate from each other at relative speed of 2u.

Perfectly Inelastic “Jousting” Collisions

canonicalD

  • For all the above,
    • the total momentum is conserved.
    • when one mass is much larger than the other, the velocity of the heavier mass is hardly changed by the collision.
    • The two masses travel at a common speed after the collision.

 

3-3-2 Types of Collisions

  • All collisions are momentum conserving. But some collisions retain more KE than others.
  • A (perfectly) elastic collision is one that retains 100% of its initial total KE.
  • A perfectly inelastic collision is one that retains the minimum amount of KE.

collisionsAfast

  • Depicted above are just 5 possible outcomes of head-on collisions of two equal masses m with equal initial speed u.
  • Notice that the total momentum (of zero) is conserved for all collisions.
  • At the top is the (perfectly) elastic collision.
  • At the bottom is the perfectly inelastic collision.

collisionsBfast

  • Depicted above are just 5 possible outcomes of head-on collisions of two equal masses m, one of them with initial speed of u, and the other initially at rest.
  • Notice that the total momentum (of mu rightward) is conserved for all collisions.
  • At the top is the (perfectly) elastic collision.
  • At the bottom is the perfectly inelastic collision. (Losing all its initial KE is impossible because of momentum conservation)

3-3-1 Principle of Conservation of Momentum

PCOM

pcom.gif

  • According to Newton’s third law, forces must come in equal but opposite pairs.
  • By extension, impulses and changes in momentum must also occur in equal but opposite pairs.
  • This means that total momentum of a system must be conserved in the absence of a net external force.

Recoil

Principle of conservation of momentum is the reason behind recoils. During the firing of the bullet, we have the rifle and the bullet pushing each other. As far as the bullet-rifle system is concerned, this is a pair of internal forces. Without any external force acting on the bullet-rifle system, its total momentum remain unchanged. Since we started with the rifle and bullet both at rest, the total momentum must remain at zero.

The bullet, though small in mass, does carry a large forward momentum thanks to its speed. This forward momentum must be matched by a backward momentum of the same magnitude.

Jet Propulsion

The principle of conservation of momentum is also the working principle behind jet propulsion. Whatever momentum gained by the exhaust in the backward direction must be matched by the momentum gained by the vehicle in the forward direction.

Explanation at xmdemo.wordpress.com/120

Explanation at xmdemo.wordpress.com/122

3-2-3-1 Impulse and Change in Momentum

Same impulse. But the collision with the wall is a stiff and snappy one: large F small Δt. Whereas the collision with the sheet is a soft and draggy one: small F large Δt. Nevertheless, the resulting Δp is exactly the same.

Bending the knees upon landing lengthens the impact duration significantly, resulting in significantly smaller F. Do realize that the impulse is not reduced. It is just spread over a longer duration.

Check out the timers at the bottom left corner. Notice how quickly the car body comes to a rest compared to the passenger. Air-bag or not, the impulses experienced by the the passenger are the same. But hitting the air bag is a lot less forceful than hitting the rigid steering wheel because of the longer impact duration.

Ah, the sight of cracked shells and spilled yoke. See if you can explain the following demonstration. 

Explanation at xmdemo.wordpress.com/121

3-2-2-1 F=dp/dt

When you’re hit by a water cannon, you’re continuously bombarded by godzillion number of H2O molecules at any one instant. The force exerted by each molecules is tiny and fleeting (lasts only for an instant). But collectively, they exert a large and continuous force.

To calculate the force exerted by a water cannon, we turn to calculating the force experienced by the water cannon instead. When the water hits you, it loses momentum. The rate at which it is losing (or changing) momentum is equal to the force that you are exerting on the water. By N3L, the water is exerting an equal but opposite force on you. Ouch.

3-2-2 Fave = Δp/Δt

During the collision, the bouncy ball’s momentum changed from mv to –mv, corresponding to a momentum change of 2mv (rightward).

The lazy ball’s momentum changed from mv to 0, corresponding to a momentum change of only mv (rightward).

Assuming that the duration of collision was about the same, this implies that the tower pushed the bouncy ball (rightward) harder than it pushed the lazy ball. Because Fpt.

By N3L, the tower was pushed harder (leftward) by the bouncy ball than by the lazy ball.

313 Newton’s Pendulu

If ONE BALL rams into the stack, why must ONE BALL come out the other end at the same speed? Why can’t TWO BALLS come out at HALF the speed, for example? (This outcome also conserves the total momentum!) And how about all FIVE BALLS swinging away at 1/5 the speed?

True enough. There are infinite number of outcomes that fulfil the Principle of Conservation of Momentum! However, since these collisions are elastic, the total kinetic energy should be unchanged after the collision as well. With this additional constraint, there is only one possible outcome. The same number of balls always come out the other end at the same end, conserving both total momentum and keeping total kinetic energy unchanged.

Let’s now bring Newton and PSY together for an xmphysics-style pendulum.

308 PCOM

pcom.gif

  • According to Newton’s third law, forces must come in equal but opposite pairs.
  • By extension, impulses and changes in momentum must also occur in equal but opposite pairs.
  • This means that total momentum of a system must be conserved in the absence of a net external force.