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Momentum and Impulse An object undergoing translational motion possesses linear momentum, p, a vector quantity, computed by taking the product of the mass and the velocity of the object. p The direction of the momentum vector is the same as that of the velocity. The Conservation of Momentum Momentum is a useful concept. In any mechanical system composed of mutually interacting objects, though not subject to external forces, the net linear momentum remains unchanged. This is the law of the conservation of momentum. The law has two mathematical forms describing the two types of collisions which objects within such a system can have with each other, elastic or inelastic collisions. In an elastic collision, the total translational kinetic energy, not only the momentum, of the colliding objects remains unchanged before and after contact; none of the kinetic energy is transformed into other types of energy, such as heat or vibrational energy. after collision p1i p1f p2i + p1i p2f p1f + p2i = pi tot = before collision total momentum remains the same before and after collision. p2f = pf tot The mathematical expression to describe conservation of momentum is as follows: Because also the kinetic energy is conserved in an elastic collision, the following expression also describes an elastic collision: © J.S Wetzel, 1993 An inelastic collision is characterized by a decrease in the translational kinetic energy of the objects upon collision. Some kinetic energy is transformed by the collision into other forms of energy such as heat or the energy of sound waves. As with an elastic collision, p1i p2i + p1i p2i = pi tot = before collision total momentum remains the same before and after collision. after collision p(1+2)f p(1+2)f momentum is conserved in inelastic collisions, though not kinetic energy. A perfectly inelastic collision is one in which the objects stick together: Because the objects stick together, for a perfectly inelastic collision you need only one equation to derive final from initial velocities. Collisions in two dimensions The illustration below shows two objects colliding elastically within a two dimensional system. Prior to the collision, object 1 possesses all of the momentum in the system. The collision enables the transfer of some of this momentum to object 2. As can be seen in the before collision p2f after collision p1f p2i = 0 p1i p1f p2f p1i = ptot lower right corner of the illustration, the vector sum of the total linear momentum in the sys- Newton's second law in terms of momentum We are all familiar with Newton's second law. An object's acceleration increases with the applied force. The acceleration diminishes, however, as the same force is applied to a more massive object. Momentum, p, is the product of the mass and the velocity. The rate of change of the velocity is the acceleration. How might we express the rate of change of the momentum? We know that force equals the mass times the acceleration or, in other words, the mass times the rate of change of the velocity. It makes sense that another wave to say force is the rate of change of the momentum. To see this better, let's use calculus. Though it won't be on the MCAT, calculus is always there. Acceleration is the rate of change of the velocity with time, or you can say that acceleration is the first derivative of the velocity a= dv dt Newton's second law can be restated with a derivative: ΣF = m dv dt Because the momentum, p, is the product of the mass and the velocity, the mass times the rate of change of the velocity equals the rate of change of the momentum: m dv dp = dt dt Therefore, Newton's second law takes an even more simple form: The net force upon an object equals the rate of change of the momentum: dp ΣF = dt Impulse Keeping in mind that the force as the rate of change of momentum, lets explore the concept of impulse, which is the product of a force and the duration. We just showed that force is the rate of change of momentum. If you multiply a rate of change by a duration of change, you will get an amount of change. The impulse is the product of the force and the time of action. It is the amount of momentum a certain force will produce over time. © J.S Wetzel, 1993