**Unit 12: Momentum and
Collisions**

**Problem Set B**

**Overview:**

Problem Set B targets your understanding of momentum conservation and your ability to apply it to the analysis of a collision or explosion.

**The Momentum Conservation
Principle**

In a collision between two objects, each object is interacting with the other object. The interaction involves a force acting between the objects for some amount of time. This force and time constitutes an impulse and the impulse changes the momentum of each object. Such a collision is governed by Newton's laws of motion; and as such, the laws of motion can be applied to the analysis of the collision (or explosion) situation. So with confidence it can be stated that ...

In a collision between object 1 and object 2, the force exerted on
object 1 (F_{1}) is equal in magnitude and opposite in
direction to the force exerted on object 2 (F_{2}). In
equation form:

The above statement is simply an application of Newton's third law
of motion to the collision between objects 1 and 2. Now in any given
interaction, the forces which are exerted upon an object act for the
same amount of time. You can't touch another object and not be
touched yourself (by that object). And the duration of time at which
you act upon that object is the same as the duration of time at which
that object acts upon you. Touch the wall for 2.0 seconds, and the
wall touches you for 2.0 seconds. Such a contact interaction is
mutual; you touch the wall and the wall touches you. It's a two-way
interaction - a mutual interaction; not a one-way interaction. Thus,
it is simply logical to state that in a collision between object 1
and object 2, the time during which the force acts upon object 1
(t_{1}) is equal to the time during which the force acts upon
object 2 (t_{2}). In equation form:

The basis for the above statement is simply logic. Now we have two equations which relate the forces exerted upon individual objects involved in a collision and the times over which these forces occur. It is accepted mathematical logic to state the following:

and **C = D **

then **A • C = - B •
D**

The above logic is fundamental to mathematics and can be used here to analyze our collision.

and **t _{1} = t_{2}
**

then **F _{1} • t_{1} =
- F_{2} • t_{2}**

The above equation states that in a collision between object 1 and
object 2, the impulse experienced by object 1 (F_{1} •
t_{1}) is equal in magnitude and opposite in direction to the
impulse experienced by object 2 (F_{2} • t_{2}).
Objects encountering impulses in collisions will experience a
momentum change. The momentum change is equal to the impulse. Thus,
if the impulse encountered by object 1 is equal in magnitude and
opposite in direction to the impulse experienced by object 2, then
the same can be said of the two objects' momentum changes. The
momentum change experienced by object 1 (m_{1} •
v_{1}) is
equal in magnitude and opposite in direction to the momentum change
experienced by object 2 (m_{2} • v_{2}).
This statement could be written in equation form as

This equation claims that in a collision, one object gains momentum and the other object loses momentum. The amount of momentum gained by one object is equal to the amount of momentum lost by the other object. The total amount of momentum possessed by the two objects does not change. Momentum is simply transferred from one object to the other object.

The logic above was used to provide an argument for that fact that in any collision between object 1 and object 2:

- the forces encountered by the objects are equal in magnitude and opposite in direction
- the times over which the interaction occurs is the same for each object
- the impulses delivered to each object are equal in magnitude and opposite in direction
- the momentum changes of each object are equal in magnitude and opposite in direction

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

- use the momentum change equation ( m
_{1}v_{1}= - m_{2}v_{2 }) with comfort and confidence. - be cognizant of units and perform unit conversions where necessary.
- recognize that a change in a quantity is calculated as the final value of the quantity minus the initial value of the quantity.
- utilize the kinematic equation d = v
_{o}•t + 0.5•a•t^{2}equation in frictionless situations in which the a term cancels. - utilize energy conservation principles to determine the height to which a pendulum rises
- 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.

View Sample Problem Set.

Return to: Set B Overview Page || Audio Help Home Page || Set B Sample Problems

Audio Help for Problem: 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 ||

Retrieve info about: Problem-Solving || Audio Help || Technical Requirements || CD-ROM

Return to: ChemPhys Problem Set Page || CP 173 Home || CP 273 Home || The Physics Classroom