Linear Momentum and Collisions

The momentum of an object is related to both its mass and its velocity. The concept of momentum leads us to a second conservation law, that of conservation of momentum. This law is especially useful for treating problems that involve collisions between objects and for analyzing rocket propulsion. In this chapter we also introduce the concept of the center of mass of a system of particles. We find that the motion of a system of particles can be described by the motion of one representative particle located at the center of mass.

Linear Momentum and Its Conservation

The General Problem-Solving Strategy and conceptualize an isolated system of two particles with masses m₁ and m₂ and moving with velocities v₁ and v₂ at an instant of time. Because the system is isolated, the only force on one particle is that from the other particle and we can categorize this as a situation in which Newton’s laws will be useful. If a force from particle 1 (for example, a gravitational force) acts on particle 2, then there must be a second force—equal in magnitude but opposite in direction—that particle 2 exerts on particle 1. That is, they form a Newton’s third law action–reaction pair, so that F₁₂=F₂₁. We can express this condition as

To finalize this discussion, note that the derivative of the sum m₁v₁ + m₂v₂ with respect to time is zero. Consequently, this sum must be constant. We learn from this discussion that the quantity mv for a particle is important, in that the sum of these quantities for an isolated  system is conserved. We call this quantity linear momentum.

The linear momentum of a particle or an object that can be modeled as a particle of mass m moving with a velocity v is defined to be the product of the mass and velocity:

p ≡ m v

If a particle is moving in an arbitrary direction, p must have three components, and Equation 9.2 is equivalent to the component equations

px =mvx                      py = mvy                      pz = mvz

Using Newton’s second law of motion, we can relate the linear momentum of a particle to the resultant force acting on the particle. We start with Newton’s second law and substitute the definition of acceleration:

 

This shows that the time rate of change of the linear momentum of a particle is equal to the net force acting on the particle.

Using the definition of momentum, Equation 9.1 can be written

where p₁f and p₂f are the initial values and p₁f and p₂f the final values of the momenta for the two particles for the time interval during which the particles interact. Equation 9.5 in component form demonstrates that the total momenta in the x, y, and z directions are all independently conserved:

This result, known as the law of conservation of linear momentum, can be extended to any number of particles in an isolated system.

Conservation of momentum

Whenever two or more particles in an isolated system interact, the total momentum of the system remains constant.

This law tells us that the total momentum of an isolated system at all times equals its initial momentum.

Impulse and Momentum

If the momentum of the particle changes from pi at time ti to pf at time tf gives

To evaluate the integral, we need to know how the force varies with time. The quantity on the right side of this equation is called the impulse of the force F acting on a particle over the time interval Δt =tf - ti . Impulse is a vector defined by

The impulse of the force F acting on a particle equals
the change in the momentum of the particle.

we see that impulse is a vector quantity
having a magnitude equal to the area under the force–time curve, as described in Figure 9.4a.
In this figure, it is assumed that the force varies in time in the general manner shown and is nonzero in the time interval

Δt =t_f – t_i

Because the force imparting an impulse can generally vary in time, it is convenient to define a time-averaged force

where Δt =tf – ti (This is an application of the mean value theorem of calculus.) Therefore, we can express Equation 9.9 as

In principle, if F is known as a function of time, the impulse can be calculated. The calculation becomes especially simple if the force acting on the particle is constant. In this case,              

 

and Equation 9.11 becomes

In many physical situations, we shall use what is called the impulse approximation, in which we assume that one of the forces exerted on a particle acts for a short time but is much greater than any other force present. This approximation is especially useful in treating collisions in which the duration of the collision is very short. When this approximation is made, we refer to the force as an impulsive force.

Collisions in One Dimension

an event during which two particles come close to each other and interact by means of forces and we can call it collision.

Perfectly Inelastic Collisions

When the colliding objects stick together after the collision, as happens when a meteorite collides with the Earth, the collision is called perfectly inelastic. Because the momentum of an isolated system is conserved in any collision, we can say that the total momentum before the collision equals the total momentum of the composite system after the collision:

Elastic Collisions

An elastic collision between two objects is one in which the total kinetic energy (as well as total momentum) of the system is the same before and after the collision. If the collision is

elastic, both the momentum and kinetic energy of the system are conserved. Therefore, considering velocities along the horizontal direction in Figure 9.9, we have

Suppose that the masses and initial velocities of both particles are known. Equations

9.15 and 9.19 can be solved for the final velocities in terms of the initial velocities

because there are two equations and two unknowns:

If particle 2 is initially at rest, then v₂i = 0, and Equations 9.20 and 9.21 become

Two-Dimensional Collisions

The game of billiards is a familiar example involving multiple collisions of objects moving on a two-dimensional surface. For such two dimensional collisions, we obtain two component equations for conservation of momentum:

Let us consider a two-dimensional problem in which particle 1 of mass m1 collides with particle 2 of mass m2 , where particle 2 is initially at rest, as in Figure. After the collision, particle 1 moves at an angle tetra with respect to the horizontal and particle moves at an angle with respect to the horizontal. This is called a glancing collision. Applying the law of conservation of momentum in component form and noting that the initial y component of the momentum of the two-particle system is zero, we obtain

the minus sign comes from the fact that after the collision, particle 2 has a y component of velocity that is downward. We now have two independent equations. As long as no more than two of the seven, we can solve the problem.

If the collision is elastic, we can also use with v₂i = 0 to give