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.

# “Classic” Collisions

Here are a few “famous” collisions whose outcomes are either obvious or can be calculated mentally.

(Perfectly) Elastic “Sitting Duck” Collisions

• 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

• 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

• 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

• 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.

• 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.

• 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

• 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.

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.

Explanation at xmdemo.wordpress.com/117

If you are sitting in a sailing boat, and there is not the slightest wind, can you get the boat moving by blowing air (with your breath) at the sail? Watch the video below to check your prediction.

Surprisingly, the outcome depends on whether the collision is elastic or not (besides the shape of the sail). Read xmdemo.wordpress.com/028 for the explanation.

# 3-3 Newton’s 3rd Law

Below are examples of action-reaction pairs. If you follow the “A-on-B and thus B-on-A” format, it is easy to identify the “reaction” force.

# 3-2-4 Impulse

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.

# 3-2-3 F=vdm/dt

The air molecules rushing out of the balloon represents a gain in (downward) momentum. The rate at which this momentum is being generated is equal to the downward force the balloon is exerting to eject the air molecules. By N3L, this is the upward thrust exerting on the balloon.

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

The change in Δp is the same for both cases. But duration of impact is much longer for the sheet. Since Fpt, the longer Δt implies a smaller average force F.

# 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

• 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.