Mechanics: Momentum and Collisions
Momentum and Collisions: Problem Set Overview
This set of 32 problems targets your ability to use the momentum equation and the impulse-momentum change theorem in order to analyze physical situations involving collisions and impulses, to use momentum conservation principles to analyze a collision or an explosion, to combine a momentum analysis with other forms of analyzes (Newton's laws, kinematics, etc.) to solve for an unknown quantity, and to analyze two-dimensional collisions. The more difficult problems are color-coded as blue problems.
An object which is moving has momentum. The amount of momentum (p) possessed by the moving object is the product of mass (m) and velocity (v). In equation form:
p = m • v
An equation such as the one above can be treated as a sort of recipe for problem-solving. Knowing the numerical values of all but one of the quantities in the equations allows one to calculate the final quantity in the equation. An equation can also be treated as a statement which describes qualitatively how one variable depends upon another. Two quantities in an equation could be thought of as being either directly proportional or inversely proportional. Momentum is directly proportional to both mass and velocity. A two-fold or three-fold increase in the mass (with the velocity held constant) will result in a two-fold or a three-fold increase in the amount of momentum possessed by the object. Similarly, a two-fold or three-fold increase in the velocity (with the mass held constant) will result in a two-fold or a three-fold increase in the amount of momentum possessed by the object. Thinking and reasoning proportionally about quantities allows you to predict how an alteration in one variable would effect another variable.
Impulse-Momentum Change Equation
In a collision, a force acts upon an object for a given amount of time to change the object's velocity. The product of force and time is known as impulse. The product of mass and velocity change is known as momentum change. In a collision the impulse encountered by an object is equal to the momentum change it experiences.
Impulse = Momentum Change
F • t = mass • Delta v
Several problems in this set of problems test your understanding of the above relationship. In many of these problems, a piece of extraneous information is provided. Without an understanding of the above relationships, you will be tempted to force such information into your calculations. Physics is about conceptual ideas and relationships; and problems test your mathematical understanding of these relationships. If you treat this problem set as a mere exercise in the algebraic manipulation of physics equations, then you are likely to become frustrated quickly. As you proceed through this problem set, be concepts-minded. Do not strip physics of its conceptual meaning.
Several of the problems in this set of problems demand that you be able to calculate the velocity change of an object. This calculation becomes particularly challenging when the collision involves a rebounding effect - that is, the object is moving in one direction before the collision and in the opposite direction after the collision. Velocity is a vector and is distinguished from speed in that it has a direction associated with it. This direction is often expressed in mathematics as a + or - sign. In a collision, the velocity change is always computed by subtracting the initial velocity value from the final velocity value. If an object is moving in one direction before a collision and rebounds or somehow changes direction, then its velocity after the collision has the opposite direction as before. Mathematically, the before-collision velocity would be + and the after-collision velocity would be - in sign. Ignoring this principle will result in great difficulty when analyzing any collision involving the rebounding of an object.
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 (F1) is equal in magnitude and opposite in direction to the force exerted on object 2 (F2). In equation form:
F1 = - F2
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 contact another object and not be contacted yourself (by that object). And the duration of time during which you contact the object is the same as the duration of time during which that object contacts you. Touch a 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 (t1) is equal to the time during which the force acts upon object 2 (t2). In equation form:
t1 = t2
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:
If A = - B
and C = D
then A • C = - B • D
The above logic is fundamental to mathematics and can be used here to analyze our collision.
If F1 = - F2
and t1 = t2
then F1 • t1 = - F2 • t2
The above equation states that in a collision between object 1 and object 2, the impulse experienced by object 1 (F1 • t1) is equal in magnitude and opposite in direction to the impulse experienced by object 2 (F2 • t2). 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 (m1 • Delta v1) is equal in magnitude and opposite in direction to the momentum change experienced by object 2 (m2 • Delta v2). This statement could be written in equation form as
m1 • Delta v1 = - m2 • Delta v2
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.
Put another way, it could be said that when a collision occurs between two objects in an isolated system, the sum of the momentum of the two objects before the collision is equal to the sum of the momentum of the two objects after the collision. If the system is indeed isolated from external forces, then the only forces contributing to the momentum change of the objects are the interaction forces between the objects. As such, the momentum lost by one object is gained by the other object and the total system momentum is conserved. And so 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.
m1 • v1 + m2 • v2 = m1 • v1' + m2 • v2'
The symbols m1 and m2 in the above equation represent the mass of objects 1 and 2. The symbols v1 and v2 in the above equation represent the velocities of objects 1 and 2 before the collision. And the symbols v1' and v2' in the above equation represent the velocities of objects 1 and 2 after the collision. (Note that a ' symbol is used to indicate after the collision.)
Momentum is a vector quantity; it is fully described by both a magnitude (numerical value) and a direction. The direction of the momentum vector is always in the same direction as the velocity vector. Because momentum is a vector, the addition of two momentum vectors is conducted in the same manner by which any two vectors are added. For situations in which the two vectors are in opposite directions, one vector is considered negative and the other positive. Successful solutions to many of the problems in this set of problems demands that attention be given to the vector nature of momentum.
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.
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 analysis which must be conducted separately. One of the analysis is a collision analysis to determine the speed of one of the colliding objects before or after the collision. The second analysis typically involves Newton's laws and/or kinematics. These two models (Newton's laws and kinematics) allows a student to make a prediction about how far an object will slide or how high it will roll after the collision with the other object.
When solving momentum plus problems, it is important to take the time to identify the known and the unknown quantities. It is helpful to organize such known quantities in two columns - a column for information pertaining to the collision analysis and a column for information pertaining to the Newton's law and/or kinematic analysis.
Habits of an Effective Problem-Solver
An effective problem solver by habit approaches a physics problem in a manner that reflects a collection of disciplined habits. While not every effective problem solver employs the same approach, they all have habits which they share in common. These habits are described briefly here. An effective problem-solver...
- ...reads the problem carefully and develops a mental picture of the physical situation. If needed, they sketch a simple diagram of the physical situation to help visualize it.
- ...identifies the known and unknown quantities in an organized manner, often times recording them on the diagram itself. They equate given values to the symbols used to represent the corresponding quantity (e.g., m = 1.50 kg, vi = 2.68 m/s, F = 4.98 N, t = 0.133 s, vf = ???).
- ...plots a strategy for solving for the unknown quantity; the strategy will typically center around the use of physics equations be heavily dependent upon an understaning of physics principles.
- ...identifies the appropriate formula(s) to use, often times writing them down. Where needed, they perform the needed conversion of quantities into the proper unit.
- ...performs substitutions and algebraic manipulations in order to solve for the unknown quantity.
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.
- Impulse-Momentum Change Equation
- Real World Applications
- Momentum Conservation Principle
- Isolated Systems
- Collision Analysis
- Explosion Analysis
- Vector Addition
- Newton's Second Law
- Kinematic Equations