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PROBLEM-SOLVING

Learn the step-by-step process for solving physics problems, including preparation, solving equations, and assessing your answers.

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PROBLEM-SOLVING

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  1. PROBLEM-SOLVING GENERAL GUIDELINES

  2. GENERAL PROBLEM SOLVING STRATEGY Solving problems require three major steps: Prepare Solve Assess

  3. PREPARE The "Prepare" step of a solution is where you identify important elements of the problem and collect information you will need to solve it. It's tempting to jump right to the "solve" step, but a skilled problem solver will spend the most time on this step, the preparation.

  4. Preparation includes: 1.Identifying the Physics Principle(s): Read the problem carefully and identify what is the underlying physics principle of the problem, then, write down the principle using an acronym such as the ones in the following list. If the problem has several steps, write down the principle(s) as appropriate. Newton's First Law (N1L) Newton's Second Law (N2L) Kinematics in One-Dimension (UAM) Kinematics in Two-Dimension (K2D) Conservation of Energy (COE) Conservation of Momentum (COM)

  5. 2.Data: Given and Unknown Make a table of the quantities whose values you can determine from the problem statement or that can be found quickly with simple geometry or unit conversions. Any relevant constants should be written here. All units should be consistent with the SI values (i.e. kg, m, s). All unit conversion should take place in this section. Also, identify the quantity or quantities that will allow you to answer the question.

  6. 3.Sketch, graph, FBD In many cases, this is the most important part of a problem. The picture lets you model the problem and identify the important elements. As you add information to your picture, the outline of the solution will take shape. If appropriate, select a coordinate axis. If the quantities involved are vectors, be sure to draw an arrow with the tip of the arrow clearly indicating the direction of the vector. When drawing a free-body-diagram (FBD), be sure to draw and clearly label only the forces acting on the system.

  7. SOLVE The "Solve" step of a solution is where you actually do the math or reasoning necessary to arrive to the solution needed. This is the part of the problem-solving strategy that you likely think of when you think of "solving problems". But don't make the mistake starting here! If you just choose an equation and plug in numbers, you will likely go wrong and will waste time trying to figure out why. The "Prepare" step will help you be certain you understand the problem before you start putting numbers in equations.

  8. Solving the problem includes: 4.Equation (always solve for unknown) Write the relevant equation or equations that will allow you to solve for the unknown. Be sure to always solve the equation for the unknown instead of just a 'plug and chug' approach. 5.Substitution Once you have solved the equation algebraically, substitute the appropriate values. 6.Answer with Units Write down the answer with the appropriate units. Remember that 'naked' numbers make no sense in Physics!

  9. ASSESS The "Assess" step of your solution is very important. When you have an answer, you should check to see if it makes sense.

  10. 7.Check the Answer  Ask yourself: - Does my solution answer the question that was asked? Make sure that you have addressed all parts of the question and clearly written down your solutions. - Does my answer have the correct units and number of significant figures? - Does the value I computed make physical sense? - Can I estimate what the answer should be to check my solution? - Does my final solution make sense in the context of the material I'm learning?

  11. MOTION An object is in motion if its position changes. The mathematical description of motion is called kinematics. The simplest kind of motion an object can experience is uniform motion in a straight line. The object experiences translational motion if it is moving without rotating.

  12. Describing Motion The study of one-dimensional kinematics is concerned with the multiple means by which the motion of objects can be represented. Such means include the use of words,graphs,equations, and diagrams.

  13. One-dimensional motion means that objects are only free to move back and forth along a single line. As a coordinate system for one-dimensional motion, choose this line to be an x-axis together with a specified origin and positive and negative directions.

  14. DISTANCE AND DISPLACEMENT Distance is the length between any two points in the path of an object. Displacementis the length and direction of the change in position measured from the starting point.

  15. DISTANCE AND DISPLACEMENT The distance an object travels tells you nothing about the directionof travel, while displacementtells you precisely how far, andin what direction,from its initial position an object is located. Distance is thetotal lengthof travel and displacement is thenet lengthof travel accounting for direction.

  16. The displacement is written: Displacement is positive. Displacement is negative.

  17. 3.1 You leave your home and drive 4.83 km North on Preston Rd. to go to the grocery store. After shopping, you go back home by traveling South on Preston Rd. a. What distance do you travel during this trip? UAM x1 = 4.83 km x2= 4.83 km distance = x1+ x2 = 4.83 + 4.83 = 9.66 km b. What is your displacement? x1= 4.83 km, N x2= 4.83 km, S displacement = Δx = x2 - x1 = 4.83 - 4.83 = 0 km

  18. SPEED If an object takes a time interval t to travel a distance x, then the average speed of the object is given by: Units: m/s 3.2 A ship steams at an average speed of 30 km/h. a. What is the speed in m/s? UAM v = 30 km/h = 8.33 m/s

  19. b. How far in km does it travel in a day? t = 24 h = 30(24) = 720 km c. How long in hours does it take to travel 500 km? x = 500 km = 16.67 h

  20. AVERAGE SPEED AND AVERAGE VELOCITY Average velocity is the displacement divided by the amount of time it took to undergo that displacement. The difference between average speed and average velocity is that average speed relates to the distance traveled while average velocity relates to the displacement.

  21. Speed: how far an object travels in a given time interval Velocity: includes directional information:

  22. 3.3 A car travels north at 100 km/h for 2 h, at 75 km/h for the next 2 h, and finally turns south at 80 km/h for 1 h. a.What is the car’s average speed for the entire journey? UAM v1 = 100 km/h t1 = 2 h v2 = 75 km/h t2 = 2 h v3 = 80 km/h t3 = 1 h Total time tT = 2 + 2 + 1 = 5 hours Total distance traveled x = v t x1 = v1t1= 100 (2) = 200 km x2 = 75 (2) = 150 km x3 = 80 (1) = 80 km xT = 200 + 150 + 80 = 430 km = 86 km/h

  23. b. What is the car’s average velocity for the entire journey? x1 = 200 km, N x2 = 150 km, N x3 = 80 km, S Displacement: 200 + 150 – 80 = 270 km Total time = 5 h = 54 km/h, N

  24. 3.4 Give a qualitative description of the motion depicted in the following x-versus-t graphs: x a. Object starts at the origin and moves in the positive direction with constant velocity. t x b. Object starts to the right of the origin and moves in the negative direction with constant velocity ending at the origin. t

  25. c. x Object starts to the right of the origin and moves in the positive direction with constant velocity. t x d. Object starts to the left of the origin and moves in the positive direction with constant velocity ending at the origin. t

  26. The slope of the graphs yields the average velocity. When the velocity is constant, the average velocity over any time interval is equal to the instantaneous velocity at any time.

  27. 3.5 Give a qualitative description of the motion depicted in the following v-versus-t graphs: a. v Object moving to the right at a slow constant speed. t b. v Object moving to the left at a fast constant speed. t

  28. ACCELERATION Acceleration is the rate of change of velocity. The change in velocity Δv is the final velocity vfminus the initial velocity vo. Acceleration happens when: An object's velocity increases An object's velocity decreases An object changes direction

  29. The acceleration of an object is given by: Units: m/s2

  30. EQUATIONS FOR MOTION UNDER CONSTANT ACCELERATION:

  31. 3.6 An object starts from rest with a constant acceleration of 8 m/s2 along a straight line. a. Find the speed at the end of 5 s UAM vo= 0 m/s a = 8 m/s2 t = 5 s = 0 + 8(5) = 40 m/s b. Find the average speed for the 5 s interval = 20 m/s

  32. c. Find the distance traveled in the 5 s = 0 + ½(8)(5)2 = 100 m or: = 20(5) = 100 m

  33. 3.7 A truck's speed increases uniformly from 15 km/h to 60 km/h in 20 s. a. Determine the average speed UAM vo= 15/3.6 = 4.17 m/s vf = 60/3.6 = 16.7 m/s t = 20 s = 10.4 m/s b. Determine the acceleration =0.63 m/s2

  34. c. Determine the distance traveled = 10.4(20) =208 m

  35. 3.8 A skier starts from rest and slides 9.0 m down a slope in 3.0 s. In what time after starting will the skier acquire a speed of 24 m/s? Assume that the acceleration is constant. UAM vo= 0 m/s, x = 9 m t = 3 s, vf = 24 m/s =2 m/s2 =12 s

  36. 3.9 A car moving at 30 m/s slows uniformly to a speed of 10 m/s in a time of 5.0 s. a. Determine the acceleration of the car UAM vo= 30 m/s vf= 10 m/s t = 5 s =- 4 m/s2 b. Determine the distance it moves in the thirdsecond = 30 (3 - 2) + ½ (- 4) (32 - 22) =20 m

  37. 3.10 The speed of a train is reduced uniformly from 15 m/s to 7.0 m/s while traveling a distance of 90 m. a. Calculate the acceleration. UAM vo= 15 m/s vf= 7 m/s x = 90 m =- 0.98 m/s2 b. How much farther will the train travel before coming to rest, provided the acceleration remains constant? =25 m

  38. 3.11 A drag racer starts from rest and accelerates at 7.40 m/s2. How far has it traveled in 1.00 s, 2.00 s, and 3.00 s? Graph the results in a position versus time graph. UAM a = 7.4 m/s2 t = 1.00 s = 3.70 m at t = 2.00 s, x = 14.8 m at t = 3.00 s, x = 33.3 m

  39. This example illustrates one of the key features of accelerated motion; positionvaries directly with the square of the time.

  40. vo = 0 x = 0 v x a t t t

  41. vo = 0 x = 0 v x a t t t

  42. vo ≠ 0 x = 0 v x a t t t

  43. vo = 0 x = 0 v x a t t t

  44. vo= 0 x = 0 v x a t t t

  45. Observation: • the sign of the velocity and the acceleration is the same if • the object is speeding up and that • the sign of the velocity and the acceleration is the opposite • if the object is slowing down.

  46. GRAPHICAL ANALYSIS OF MOTION • Graphical interpretations for motion along a straight line (the x-axis) are as follows: • the slope of the tangent of an x-versus-t graph and define instantaneous velocity, • the slope of the v-versus-tgraph and understand that the value obtained is the average acceleration, • the area under the v-versus-t graph and understand that it gives the displacement, • the area under the a-versus-t graph and understand that it gives the change in velocity.

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