Inertia Vs Momentum

Momentum: The resistance of an object to a change in its state of motion.

That sounds like a fishy definition of momentum to me. A slightly better definition, at least at your level, is that momentum represents the "amount of motion" an object has. Granted, "amount of motion" is a very vague term, but it stands to reason that if "amount of motion" were to be precisely defined, it would have the following characteristics:

• The larger an object, the larger its "amount of motion", all else being equal
• The faster an object is moving, the larger its "amount of motion", all else being equal
• If two objects have equal "amounts of motion" in opposite directions, the total "amount of motion" of the system of both of them is zero

Momentum is a precisely defined quantity that satisfies these principles. For slow-moving, massive objects, it can be calculated by $p = mv$.

A much better definition of momentum comes from the fact that it is conserved, but given the level at which you're currently studying, you probably haven't yet been taught enough background to really understand that, so I'll leave it out. (But do know that the fact that momentum is conserved in many situations is what makes it so useful.)

For inertia, on the other hand, this is quite reasonable:

Inertia: a characteristic of an object that resists changes to its state of motion.

The idea is that objects which are harder to move, or whose motion is harder to change, have more inertia. The precisely defined quantity that satisfies these properties is inertial mass, which you probably know as just "mass". (Or actually, energy, but at your level you can pretend it's just mass. You probably won't run into situations where this difference between mass and energy becomes important for quite a long time.)

Inertia is an intrinsic characteristic of the object related to its mass. Inertia tells you how much force it will take to cause a particular acceleration on the object.

Momentum is a function of an object's mass and velocity. Momentum is a measure of the kinetic energy of the object.

A massive object can have any momentum (at least as long as its velocity is less than light speed) including zero or negative momentum depending on the reference frame and coordinate conventions, but always has positive nonzero inertia.

Good question. Clear thinking is essential in physics, and too many teachers and students just move forward to plug-and-chug equations without first spending enough time to study, explain, and fully understand the behaviors behind the math.

Short Answer:

Inertia is a general concept that describes the observed behavior of objects in motion and at rest that Newton wrote out in his first law. It has no magnitude or measurement units. To quantify Inertia, physicists developed the more specific concept of Momentum, which has quantities of mass, speed, and direction.

Long Answer:

Inertia is the observed natural tendency of an object in motion to keep moving in the same direction and at the same speed, or if at rest, to stay at rest. In physics, we quickly learn the the only difference between an object moving or being at rest is the relative motion of the observer, so both of these cases are really the same thing. Inertia simply tells us that to change the motion of an object requires applying a force to it.

To fully describe the property of Inertia with units we can measure, we must quantify an object's speed of motion, direction of motion, and resistance to change of motion. We combine these three things into a single parameter for each object called its Momentum.

The resistance to change of motion does not depend upon direction, so it is what we call a "scalar" quantity, and has been named Mass. We quantify Mass using units such as kg or lb-mass. We actually measure the Mass of an object indirectly by applying a known Force to it and measuring its change in motion (acceleration) according to the equation Mass = Force / Acceleration. Objects that require a lot of Force to accelerate a little bit have large Mass. Objects that accelerate a lot with a little bit of Force have small Mass.

Speed is also a scalar quantity, but when we combine speed with direction, we get a "vector" quantity called Velocity.

Momentum has both a magnitude and a direction. We quantify the Momentum of an object as the product of the Mass scalar quantity times its Velocity vector.

Momentum is also a measurable property of a sets of objects. Their individual Momenta can be added together using vector addition and be represented by a virtual object we call a Center of Mass moving with a Net Velocity.

Momentum is "conserved", which simply means that it does not change over time for any closed system unless some external force is applied. For a collection of objects, their collective Momentum does not change, even if they bang into each other and bounce apart again or clump together, or one object goes spinning off away from its neighbors. These collisions, if not perfectly elastic, will reduce the Kinetic Energy of the system, especially if they clump together into a single object, but the Momentum of the clump will be the same as the net momentum of all the original individual pieces -- the Center of Mass will continue to move with the same Net Velocity. By this example we can see that we should not confuse Momentum (Mass x Velocity) with Kinetic Energy (Mass x Velocity^2). While Total Energy is conserved (first law of thermodynamics), Kinetic Energy can shift into other forms such as Thermal Energy or Potential Energy, so it is not conserved like Momentum.

Note:

As an important caveat, the above applies to the realm of classic Newtonian physics where velocities are much, much less than the speed of light (speed of information) and relativistic effects are negligible. As objects approach the speed of light, velocity and mass are not so distinct, and additional energy appears to the observer to increase the mass of an object rather than its velocity. Still, momentum is conserved.