Arizona State University| PHY 121: Univ Physics I: Mechanics|Homework 1-1| General Problem Solving Strategy

Learning Goal:

To practice Problem-Solving Strategy for general problems.

GENERAL PROBLEM-SOLVING STRATEGY

MODEL:It is impossible to treat every detail of the situation. Simplify the problem with a model that captures the essential features. For example, the object in a mechanics problem is usually represented as a particle.

VISUALIZE:This is where expert problem solvers put most of their effort.

  • Draw a pictorial representation. This helps you visualize important aspects of the physics and assess the information you are given. It starts the process of translating the problem into symbols.
  • Use a graphical representation if it is appropriate for the problem.
  • Go back and forth between these representations; they need not be done in any particular order.

SOLVE:Only after modeling and visualizing are complete is it time to develop a mathematical representation with specific equations that must be solved. All symbols used here should have been defined in the pictorial representation.

ASSESS:Is your result believable? Does it have proper units? Does it make sense?

PROBLEM

Two hockey pucks, labeled A and B, are initially at rest on a smooth ice surface and are separated by a distance of 18.0 m. Simultaneously, each puck is given a quick push, and they begin to slide directly toward each other. Puck A moves with a speed of 2.90 m/s, and puck B moves with a speed of 3.90 m/s.

A) Which of the motion diagrams shown here best represents the motion of puck A prior to the collision?

The figure shows six motion diagrams. Diagram A shows 0 acceleration and equal segments. Diagram B shows a forward acceleration and lengthening segments. Diagram C shows a backward acceleration and shortening segments. Diagram D shows a forward acceleration and equal segments. Diagram E shows 0 acceleration and shortening segments. Diagram F shows a forward acceleration and rapidly lengthening segments.

ANSWER: A

B) After completing your motion diagram, it’s time to choose an appropriate coordinate system for this problem. Note that there is no single correct way to visualize this problem, but for the questions that follow, assume that the pucks are moving along the x-axis with +x directed to the right. Take the initial position of puck A, when it is at rest, to be at the origin, and take the initial position of puck B to be to the right of puck A. Start your clock at the instant when the pucks begin to move. Based on the assumptions listed above, list the quantities as known or unknown.

ANSWERS:

KNOWN:  the initial position of puck A; the initial position of puck B; the initial time (when the pucks begin to move); the initial velocity of puck A; the initial velocity of puck B

UNKNOWNS: the moment of collision; the position of puck A at the moment of collision; the position of puck B at the moment of collision

C) Which of the following relationships follows from the problem statement? Note that (x1)A and (x1)B are the positions of the pucks at the moment of their collision.

ANSWER: \left(x_1\right)_A=\left(x_1\right)_B

For the pucks to collide, they must be at the same position at the same time! So, now the number of unknown quantities in this problem is reduced to two: the final position of the pucks, which you may simply call x_1, and the time of collision, t_1. Make certain to identify what the problem is trying to find.Now, put all this information together and create your pictorial representation for this problem. Your effort should produce a sketch like this:The figure shows a motion diagram and a sketch of a head-on collision of two pucks. The motion diagram shows the pucks moving toward one another with zero acceleration. Three segments of equal lengths correspond to each puck. The sketch shows four positions of the pucks, from their initial positions at time t subscript 0 baseline to the common final position at time t subscript 1 baseline.A complete pictorial representation would also include a list of knowns and unknowns similar to the following: 

\left(x_0\right)_A=0\:m,\:\left(x_0\right)_B=18.0\:m

Known: (v0x)A= 2.90 m/s , (v0x)B= 3.90 m/s , t0=0aA=aB=0

Unknown: t_1,\:\left(x_1\right)_A=\left(x_1\right)_B=x_1

Note that your target variable is the distance covered by puck A by the time the two pucks collide, that is, \left(x_1\right)_A-\left(x_0\right)_A. Since we selected a coordinate system whose origin is at the initial position of puck A, \left(x_0\right)_A=0\:m and your target variable becomes simply x_1.In addition to the pictorial representation, you could also draw a graphical representation of the problem, such as plotting (on the same graph) the position of each puck as a function of time.

D) What is x_1, the distance that puck A covers prior to the collision?

ANSWER: 7.68 m

In the previous step you determined that x1=x1A=x1B. Combine this information with the equations you were given. You will then obtain the following systems of equations: 

x_1=\left(x_0\right)_A+\left(v_{0_x}\right)_At_1

x_1=\left(x_0\right)_B+\left(v_{0_x}\right)_Bt_1

Solve this system to find an expression for x1 in terms of known quantities. Then substitute the values of the known quantities into your expression to find a numerical result. The resulting expression should be

x_1=\frac{x_{0B}+\frac{v_{0_B}x_{0_A}}{v_{0_A}}}{1+\frac{v_{0_B}}{v_{0_A}}}

If you substitute the given values, you must come up with

x_1=7.68\:m

E) Your answer should satisfy common sense. For instance, can you decide which of the following values for the distance covered by puck A would definitely be wrong, regardless of the speed of the two pucks and considering that the two pucks are sliding toward each other?

  • 19 m

  • 5 m

  • 16 m

  • 1 m

  • 25 m

ANSWER: AE

Even without solving the problem, we can conclude that the distance covered by puck A must be less than or equal to 18.0 m . You can verify that your answer from Part D does satisfy this common-sense evaluation.

In the future, whenever possible, use two simple methods, units check and common-sense evaluation, to assess your results. They will be useful in catching some incorrect answers without following the entire solution through. However, beware that these methods do not catch all incorrect answers.

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