Physics 163

Unit 4: 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 problem set will test your ability to read carefully and identify known and unknown information and to interpret wording regarding inelastic and elastic collsions.

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.

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).

A detailed description of the momentum conservation principle is provided online at The Physics Classroom.

Solving Momentum Conservation Problems

The momentum conservation problems in this problem set describe a collision between two objects. In the problem description, a variety of known quantities are explicitly stated or implied. One of the variables in the momentum conservation equation is not known and the problem requests that the solver determine the value of that variable.

The problem-solving process requires that you carefully read the problem, visuialize the situation and determine all known numerical values. To assist in the visualization process, a before- and after-collision diagram is often drawn. An example is shown below.

During the reading process, you will need to extract numerical information and associate the known values with the symbols used in the above equation. This is often done on the before- and after-collision diagram as shown above. If a clause within the problem indicates that one of the objects was are rest prior to the collision, then you can assign 0 m/s as the pre-collision velocity value for that object. If a clause within the problem indicates that the objects move together as a single unit after the collision, then you will know that v1' = v2' and you can represent the two variables by the single variable v'. Finally, when reading a problem description you should pay careful attention to direction information. Momentum is a vector and has a direction associated with it; the direction of the momentum vector is the same as the direction of the velocity vector. Mathematically, this direction is often represented by a positive or negative value. For instance, if object 1 is moving to the right, then its velocity is assigned a + value; and if object 2 is moving to the left, then its velocity is assigned a - value. These positive and negative numerical values must ultimately be substituted into the momentum conservation equation.

Once the problem is read carefully and the known numerical information is identified, you will need to identify the unknown quantity. Once identified, a strategy should be plotted which includes the use of the momentum conservation equation to solve for the unknown quantity.

Using a Momentum Table to Solve Problems

A convenient way to approach a momentum problem involving two objects is to utilize a table in which you keep track of the before- and after-collision of the objects. Let's consider how this approach can be used for the following sample problem.

A blue car with a mass of 1000 kg is moving at 8.0 m/s collides with a 12000-kg red car which is at rest. The two cars stick together and move as a single unit after the collision. Determine the post-collision speed of the cars.

In a momentum table, identify the two objects in separate rows and calculate their momenta in two columns as shown below. If a quantity such as mass or post-collision velocity is unknown, then represent it by a symbol such as m or v' in the cell of the table. Use a thrid raw in which you represent the total momentum of the system by summing the two rows above it.

 Before Collision After Collision Blue Car 1000 kg • 8 m/s = 8000 kg m/s 1000 kg • v' Red Car 1200 kg • 0 m/s = 0 kg m/s 1200 kg • v' Total for System 8000 kg m/s 1000 kg • v' + 1200 kg • v'

Once you have completed the table, equate the before- and after collision momentum of the system to each other and utilize algebra to solve for the unknown quantity as shown below.

8000 kg m/s =1000 kg • v' + 1200 kg • v'

8000 kg m/s = 2200 kg • v'

(8000 kg m/s) / (2200 kg) = v'

3.6 m/s = v'

A detailed description of the use of a momentum table to solve collision problems is provided online at The Physics Classroom.

View Sample Problem Set.

 Problem Description Audio Link 1 Calculation of final velocity for an inelastic collision 2 Calculation of final velocity for an inelastic collision 3 Calculation of mass of an object an inelastic collision; similar to a lab done in class 4 Calculation of final velocity for an inelastic collision 5 Calculation of initial velocity for an inelastic collision; must give attention to vector nature of momentum 6 Calculation of final velocity for an inelastic collision; must give attention to vector nature of momentum 7 Calculation of final velocity for an inelastic collision 8 Calculation of final velocity for a situation in which an impulse causes a separation of two moving objects 9 Explosion problem; must pay attention to units; similar to a lab done in class 10 Calculation of final velocity for an elastic collision; must give attention to vector nature of momentum 11 Calculation of final velocity for an elastic collision 12 Use of equations as a guide to thinking; proportional reasoning technique 13 Explosion problem; must pay attention to units and be very organized 14 Use of equations as a guide to thinking; proportional reasoning technique

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