Momentum
From MyMCAT
Contents |
Introduction
Momentum in essence is a measure of how hard it is to stop a moving object; The faster an object is moving or the heavier it is, the hard it is to stop it that object. More accurately however, momentum is a vector and can be expressed as
Intuitively, this matches the basic definition as both mass and velocity are involved. The direction of the momentum is in the same direction as the velocity of the object. (Note that P is the symbol for momentum as m is used already.)
Conservation of Momentum
Momentum is conserved in any collision if the effect of any external forces present is negligible relative to the effect of the collision. Thus in general, we can assume that the sum of all the momentums before a collision must equal the sum of the momentums after.
In the above example we have two objects each with their own momentum, m1v1 and m2v2. As a result of the conservation of energy, the sum of these two momentums will equal the sum of the two momentums after. Or,
While there are a lot of variables, generally all but one will be given and a simple substitution and solving for the remaining term will yield what you are trying to determine. Often the formula can be simplified if masses are equal or some objects are stationary. (An object with no velocity has no momentum.)
Momemtum is also conserved in two dimensions. Thus, any two dimensional problem and be looked at in its components. Consider the collision below,
At first glance this problem may seem difficult, however no new equations are necessary to solve the system. The momentum in the x direction before and after the collision must be conserved and the momentum in the y direction before and after must be conserved. In this example, if we consider the x direction, we have one large object moving to the right, thus after the collision both will be moving to the right, but neither with as much speed as the original velocity. In the y direction there is no momentum initially, thus, after the collision, the momentum must remain at zero but because we know the objects are moving up and down, the two momentums must be equal and opposite.
Conservation of Energy
While this section does not cover energy in its full terms, collisions do involve energy and energy may or may not be conserved. If during a collision energy is conserved, it is said to be a perfectly elastic collision. In perfectly elastic collisions the total kinetic energy is also conserved and a second formula becomes relevant,
In an inelastic collision, energy is not conserved thus the kinetic energy before must be greater than the kinetic energy after. In the figure above, energy is lost as a result of the explosion and thus it is an inelastic collision. A special case of this is a perfectly inelastic collision in which the two objects stick together once they collide. In this case, the formula for momentum reduces to:
In this example of a perfectly inelastic collision, the two objects become fused during the collision and thus act as one larger object after.
But again, in both cases of inelastic collisions we cannot use the second formula as energy is not conserved. Where did this energy go? Well if we think of cars crashing, generally the energy is lost in causing the metal of the cars to rip and tear apart, glass shattering, the bang sound of the crash, and any other process that involves energy. In real life there are very few examples of elastic collisions as most lose some energy, even the example of billiard balls is not perfect, but for the most part we can assume it is close enough.
