Physci text 04

Conservation of momentum and conservation of energy

Momentum

While momentum is mathematically defined in physical science as the mass of an object multiplied by the velocity of the object, momentum can also be thought of as being the tendency of an object in motion to remain in motion. An object at rest will tend to remain at rest. 

Momentum investigation questions

• When one marble collides with a line of five marbles, what happens?

• When two marbles collide with a line of five marbles, what happens?

• For how many inbound marbles does a pattern continue?

• When the inbound marbles are rolled in slowly, what happens?

• When the inbound marbles are rolled in quickly, what happens?

Sir Isaac Newton wrote in a letter to the Jesuit scientist Ignace Gaston Pardies:

For the best and safest method of philosophising seems to be, first to enquire diligently into properties of things, and to establish those properties by experiments and then to proceed more slowly to hypotheses for the explanation of them. For hypotheses should be employed only in explaining the properties of things, but not assumed in determining them; unless so far as they may furnish experiments.

Letter to the French Jesuit, Gaston Pardies. Translation from the original Latin, as in Richard S. Westfall, Never at Rest: a Biography of Isaac Newton (1983), page 242. Direct source: The Renaissance Mathematicus

Newton started with diligent inquiry using experiments to establish properties and then used those properties to build hypotheses. The Greek philosophers, in contrast, often started with a hypothesis and rarely did experiments. In this exploration you are starting with inquiry and attempting to move towards a hypothesis.

Post-exploration considerations

This exploration was of conservation of momentum in a collision of marbles. Conservation means "stays the same." In a momentum conserving collision we mean that the momentum after an event is the same as the momentum before an event. For this exploration the "event" was a collision between sets of marbles.

Momentum is the mass multiplied by the velocity. The lowercase Greek letter rho ρ is used for momentum, m is used for mass, and v is used for velocity (speed). The lower case Greek letter ρ looks like the English letter "p" but is the lowercase letter "r" in Greek. We can see this ρ shape in the English capital letter R. I have never seen an explanation of why density uses the letter ρ, but "d" was already in use for distance. One idea, unproven, is that ρ came from considerations of volumetric flow rate for fluids, the "r" coming from the "r" in rate.

The momentum ρ is defined as the mass multiplied by the velocity (speed). Momentum takes into account both the mass and the velocity. If "mass in" equals "mass out" and "speed in" equals "speed out," then the product of mass times speed in should also be equal to mass times speed out. That is, "momentum in" should be equal to "momentum out." This is called conservation of momentum.

Both momentum and velocity have directions associated with them, both are vector quantities. This means they are usually written with an arrow on top of the symbol for them. Marbles have a mass, their velocity is a speed in a particular direction. The tracks keep the marbles moving in the same single direction. In the world of science this is a one-dimensional model and keeps the mathematics simpler.

Written without the vector notation and subscripts, the momentum is ρ = mv where m is the mass and v is the velocity.

Conservation of energy

Energy in this section refers to two different types of energy: gravitational potential energy and kinetic energy. Gravitational potential energy is the energy of position. Kinetic energy is the energy of motion.

Laboratory four explores the relationship between the gravitational potential potential energy and kinetic energy. Gravitational potential energy is energy due to an elevated position. Kinetic energy is the energy an object has due to motion. When a marble rolls down a slope the gravitational potential energy is converted to kinetic energy. The same happens when a car rolls down a hill.

Gravitational Potential energy

Gravitational potential energy is energy contained in an object due to its position or composition. Objects at rest a height h above a surface have gravitational potential energy due to their position. Gravitational potential energy is equal to the mass multiplied by the acceleration of gravity g multiplied by the height.

Gravitational potential energy = mass × gravity × height

Note that the acceleration of gravity g is 980 cm/s². Although on Pohnpei the acceleration of gravity is 979 cm/s², that value is often rounded off to 980 cm/s², a value closer to the global average of 980.7 cm/s².

The "potential" part of the phrase refers to the "potential" for the energy to be converted to other forms of energy. In this activity the gravitational potential energy will be converted to kinetic energy. Gravitational potential energy is not the only type of potential energy, there are other forms of potential energy. Springs and rubber bands can store potential energy, as can bonds between atoms.

Letters in physical science can be very confusing. The g above is being used as a variable. In physical science g as a variable means the acceleration of gravity. The letter "g" is also used as unit. The letter g as a unit means "grams." Grams is a measure of the mass m. A regular marble has a mass of about five grams, often written as 5 g. Letters in physical science are often used with different meanings as variable and units. Only the context can tell you the meaning of a letter.

Kinetic energy

Kinetic energy is the energy of motion, the energy that an object with a mass m has as a result of moving at a velocity v. Kinetic energy is actually the integration of the momentum with respect to velocity, but that is mathematics that is beyond the scope of this course. The kinetic energy KE can be calculated from the equation ½mv²

Conservation of energy

The Conservation of energy states that energy can neither be created nor destroyed. Energy can only be converted between different types of energy. The conservation of energy suggests that the gravitational potential energy of a marble on a slope should be equal to the kinetic energy of the marble at the bottom of the slope.

There is one complication in the following calculation: at the bottom of the slope the marble has TWO kinds of kinetic energy. Have you ever spun a coin on one edge? The coin keeps spinning. The spinning coin has rotational kinetic energy. There is also rotational momentum. Thus the following includes consideration of both the linear kinetic energy of the marble in motion and the rotational energy of the marble because the marble is also spinning as the marble rolls.

Instructional note for instructors: What follows is the theory. The key point here is not that students will derive these equations, but to show how the mathematical equation is derived. The students are not trying to construct the theory.

The conservation of energy can be expressed mathematically as:

gravitational potential energy due to vertical height = kinetic energy of the marble at the bottom of the slope

Remember that there are TWO forms of kinetic energy to account for in this experiment: linear kinetic energy and rotational kinetic energy. Therefore:

GPE = KE + RE.

Gravitational potential energy = mass m × acceleration of gravity g × height h.

Linear kinetic energy (KE) is the energy of an object moving across a surface. Linear kinetic energy is equal to:

KE = ½mv²

where m is mass and v is the linear velocity.

Rotational kinetic energy (RE) is the energy of a rotating object. Rotating means spinning. As the marble rolls the marble spins. Rotational kinetic energy is equal to:

RE = ½Iω²

..where I is the rotational inertia of the spinning object and omega ω is the rate of rotation in unitless radians per second. ω is equal to v÷r

The following is a sketch of simplifying GPE = KE + RE.

KE + RE = GPE
½mv² + ½Iω² = mgh
½mv² + ½(0.4mr²)(v÷r)² = mgh
divide through by mass (mass does not change the effect of the acceleration of gravity!
½v² + ½(0.4r²)(v÷r)² = gh
cancel the r² in the rotational inertia expression numerator and denominator
½v² + ½(0.4)(v)² = gh
0.5v² + 0.2v² = gh
0.7v² = gh
v² = gh÷0.7
v = √(gh÷0.7)
Take g to be 980 cm/s² (the acceleration of gravity)
v = √(1400h)
v = √1400 ×√h
v = 37.4√h

In the above expression 1400 is in the units of acceleration: cm/s². After substitution and simplification the relationship between the velocity and the height is given by the following equation:

v = 37.4√h

Note that the mass and radius (size) of the marble is irrelevant to the speed. Note too that the speed is proportional to the square root of the height. This is not a direct relationship, not a linear relationship, this is a square root relationship. A parabolic on it's side.

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