**Unit 12: Momentum and
Collisions**

**Problem Set D**

**Overview:**

Problem Set D targets your ability to use momentum principles to analyze a collision. Three types of difficult problems comprise the majority of this problem set. The three types of difficult problems are often described as

- momentum plus problems
- elastic collision problems
- two-dimensional collision problems

Each of these problem types are described below.

**The Momentum Conservation
Principle**

When a collision occurs between two objects in an *isolated
system*, the total system momentum is conserved. A system is
considered isolated if the only forces contributing to a momentum
change for an individual object are the interaction forces acting
between the colliding objects. If total system momentum is conserved
when objects 1 and 2 collide together, then the total momentum of the
system before the collision is equal to the total momentum of the
system after the collision. That is, the sum of the momentum of
object 1 and the momentum of object 2 before the collision is equal
to the sum of the momentum of object 1 and the momentum of object 2
after the collision. The following mathematical equation is often
used to express the above principle.

The symbols **m _{1}** and

**Elastic Collisions**

A collision between two objects is considered elastic if the total amount of kinetic energy (system KE) is conserved during the collision. That is, an elastic collision is a collision in which the total amount of kinetic energy before the collision is the same as the total amount of kinetic energy after the collision. Put in equation form:

**0.5 • m _{1} •
v_{1}^{2 } + 0.5 • m_{2} •
v_{2}^{2 } = 0.5 • m_{1} •
(v_{1}')^{2 } + 0.5 • m_{2} •
(v_{2}')^{2} **

To be truly elastic, a collision would have to occur between objects due to repulsive forces acting from some distance away. When objects touch and make contact upon collision, a certain amount of mechanical energy is converted to non-mechanical forms - sound, heat and light. Such a loss of energy to non-mechanical forms would result in a loss of total system kinetic energy. So ideally, an elastic collision occurs when the forces causing the collsion are non-contact forces such as electric repulsion or magnetic repulsion forces. Many collisions are approximated as elastic collisions because the amount of loss of kinetic energy is so small that it is nearly negligible. Though not proven here, the above kinetic energy conservation equation can be simplified into the following for elastic collisions in isolated systems:

A difficult elastic collision problem is one in which there are two unknowns - such as the post-collision velocity of both objects. To solve such an equation for two unknowns, it is necessary to have two independent equations. One equation is the usual momentum conservation equation; the second equation is the simplified form of the kinetic energy conservation equation. Once two equations are generated for the two unknowns, a collection of algebraic steps must be performed to solve for the unknown(s).

**Momentum Plus Problems**

A momentum plus problem is a problem type in which the analysis and solution includes a combination of momentum conservation principles and other principles of mechanics. Such a problem typically involves two or more analysss which must be conducted separately. One of the analyses is a collision analysis to determine the speed of one of the colliding objects before or after the collision. The second analysis typically involves work and energy equations or Newton's laws and kinematics. Either of these two models (work-energy and Newton's laws/kinematics) allows a student to make a prediction about how far an object will slide or how high it will roll or swing after the collision with the other object.

**Two-Dimensional Collision
Problems**

A two-dimensional collision is a collision in which the two objects are not originally moving along the same line of motion. They could be initially moving at right angles to one another or at least at some angle (other than 0 degrees and 180 degrees) relative to one another. In such cases, vector principles must be combined with momentum conservation principles in order to analyze the collision. The underlying principle of such collisions is that both the "x" and the "y" momentum are conserved in the collision. The analysis involves determining pre-collision momentum for both the x- and the y- directions. If inelastic, then the total amount of system momentum before the collision (and after) can be determined by using the Pythagorean theorem. Since the two colliding objects travel together in the same direction after the collision, the total momentum is simply the total mass of the objects multiplied by their velocity. If the collision is elastic, then an expression for momentum for each object must be written for both the x- and the y- directions.

To be successful on this problem set, you will have to:

- use the momentum conservation equation (
**m**_{1}• v_{1}+ m_{2}• v_{2}= m_{1}• v_{1}' + m_{2}• v_{2}'_{ }) with comfort and confidence. - be cognizant of units and perform unit conversions where necessary.
- analyze elastic collisions using the principles and equations described above.
- analyze a 2-dimensional collision using momentum principles and right-angle trigonometry.
- give attention to directional information and recognize that momentum is a vector.
- utilize energy conservation principles to relate a pre-collision speed to an initial height of a moving object and to relate a post-collision speed to the final height of an object.
- utilize energy conservation principles to relate the height to which a pendulum rises to its velocity at the bottom of its arc.
- perform a projectile analysis to determine the displacement (horizontally) of a projectile lauched horizontally from a table of a given height.
- practice the habits of a good problem solver.

**Additional Readings/Study
Aids:**

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems.

Momentum Conservation Principle | Isolated Systems | Collisions | Explosions

Situations Involving External Forces | Situations Involving Energy Conservation

Vector Addition | Horizontally-Launched Projectiles | Spring Energy

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